-groups related to exceptional groups of Lie type€¦ · p-groups related to exceptional groups of Lie type Saul D. Freedman, BSc (Hons), BE (Hons) October 2018 Department of Mathematics

This thesis is presented for the degree of Master of Philosophy (Research) inMathematics at The University of Western Australia

p-groups related to exceptional groups of Lie type

Saul D. Freedman, BSc (Hons), BE (Hons)

October 2018

Department of Mathematics and StatisticsSchool of Physics, Mathematics and Computing

Supervisors: Assoc/Prof John Bamberg (Coordinating)Dr Luke Morgan

i

Thesis Declaration

I, Saul Freedman, certify that:

This thesis has been substantially accomplished during enrolment in the degree.

This thesis does not contain material which has been accepted for the award of any

other degree or diploma in my name, in any university or other tertiary institution.

No part of this work will, in the future, be used in a submission in my name, for

any other degree or diploma in any university or other tertiary institution without

the prior approval of The University of Western Australia and where applicable,

any partner institution responsible for the joint-award of this degree.

This thesis does not contain any material previously published or written by

another person, except where due reference has been made in the text.

The work is not in any way a violation or infringement of any copyright,

trademark, patent, or other rights whatsoever of any person.

This thesis contains work submitted and in preparation for publication, some of

which has been co-authored.

Signature:

Date: 9/8/18

ii

Abstract

Representation theory allows us to consider an arbitrary finite group as a sub-

group of a general linear group defined over a finite field. This linear subgroup is

a group of automorphisms of an associated vector space, which is an elementary

abelian p-group. Bryant and Kovacs proved a non-abelian analogue of this fact.

Namely, if H is a subgroup of the general linear group GL(d, p), with d > 1 and p

prime, then there exists a non-abelian p-group P such that H is the group A(P )

induced by Aut(P ) on the Frattini quotient P/Φ(P ). Moreover, the action of H

on the vector space Fdp is equivalent to the action of A(P ) on P/Φ(P ). Bamberg,

Glasby, Morgan and Niemeyer showed that when H is a suitable maximal subgroup

of GL(d, p), we can choose P to be “small”, in terms of its exponent-p class, nilpo-

tency class, exponent and order. However, it is not known when this is possible in

general. Since the finite simple groups are the “building blocks” of finite groups,

considering the cases where H is related to these simple groups may help us to solve

this problem for arbitrary linear groups.

In this thesis, we consider certain cases where H contains the (finite) simply

connected version G of an exceptional Chevalley group defined over the field Fp,with p odd (and p > 3 in some cases). In most cases, H is the normaliser of G

in GL(d, p), where d is the minimal dimension of an irreducible Fp[G]-module. In

the remaining cases, H is equal to the subgroup G of GL(d, p). We construct a

p-group for each group H that is as small as possible, in terms of its exponent-p

class, exponent and nilpotency class, and in most cases, also in terms of its order.

In order to construct each p-group, we use highest weight theory to describe the

submodule structure of the exterior square or third Lie power of each irreducible

Fp[G]-module of minimal dimension, and we explore part of the overgroup structure

of G in GL(d, p). This allows us to determine the stabiliser in GL(d, p) of each

submodule of the aforementioned modules, and we use this knowledge to construct

the desired p-group as a quotient of an appropriate universal p-group. In fact, the

information here about stabilisers of submodules is of general interest, and so we

explore this representation theory over each finite field of characteristic p.

iii

Acknowledgements

This research was supported by a Hackett Postgraduate Research Scholarship,

and an Australian Government Research Training Program (RTP) Scholarship. I

would like to thank Assoc/Prof John Bamberg and Dr Luke Morgan for their in-

valuable support throughout my research, and for helping me to produce the best

work possible. I am also grateful to Professors Martin Liebeck, Donna Testerman

and Gunter Malle for helpful discussions regarding highest weight theory. Finally, I

am incredibly thankful for the support of my friends and their interest in my work.

iv

Authorship declaration: co-authored publications

This thesis contains work that has been adapted and submitted for publication.

Details of the work:

John Bamberg, Saul D. Freedman, and Luke Morgan. On p-groups with

automorphism groups related to the Chevalley group G2(p).

Location in thesis:

Sections 5.2–5.3, and some details from Chapters 3 and 6.

Student contribution to work:

Although the submitted paper has been adapted from parts of this thesis,

all work in the thesis is the student’s own.

Co-author signatures and dates:

Student signature and date:

I, John Bamberg, certify that the student statement regarding their

contribution listed above is correct

Coordinating supervisor signature:

Date:

Authorship declaration: sole author publications

This thesis contains the following sole-authored work that is being adapted and

prepared for publication.

Details of the work:

Saul D. Freedman. On p-groups with automorphism groups related to the

exceptional Chevalley groups.

Location in thesis:

Chapters 3–7.

Signature:

Date:

9/8/18

9/8/18

9/8/201810/8/18

9/8/2018

Contents

Abstract ii

Acknowledgements iii

1 Introduction 1

2 Preliminaries 5

2.1 Subgroups and quotient groups . . . . . . . . . . . . . . . . . . . . . 5

2.2 Almost simple and quasisimple groups . . . . . . . . . . . . . . . . . 9

2.3 Nilpotent groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 p-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.6 Classical groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.7 Tensor powers and exterior powers of vector spaces . . . . . . . . . . 19

2.8 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.9 Representation theory: subgroups, subfields and extension fields . . . 30

2.10 Aschbacher’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.11 Lie powers of vector spaces . . . . . . . . . . . . . . . . . . . . . . . . 38

3 Induced linear groups and universal p-groups 41

3.1 Inducing a linear group on P/Φ(P ) . . . . . . . . . . . . . . . . . . . 41

3.2 Groups of prime exponent . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3 p-covering groups and p-groups of exponent-p class 2 . . . . . . . . . 49

4 Simple groups of Lie type and highest weight theory 55

4.1 The simple groups of Lie type . . . . . . . . . . . . . . . . . . . . . . 55

4.2 Highest weight theory . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.3 Multiplicity free K[G]-modules . . . . . . . . . . . . . . . . . . . . . 66

4.4 The linear algebraic group G2 . . . . . . . . . . . . . . . . . . . . . . 69

4.5 The remaining exceptional groups of Lie type . . . . . . . . . . . . . 73

4.6 Finite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5 Overgroups of exceptional Chevalley groups 101

5.1 General results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

v

vi contents

5.2 G2(F) and the octonion algebra . . . . . . . . . . . . . . . . . . . . . 108

5.3 Overgroups of G2(q) in GL(7, q) . . . . . . . . . . . . . . . . . . . . . 112

5.4 Overgroups of F4(q) in GL(25, q) or GL(26, q) and of E8(q) in GL(248, q)113

5.5 Overgroups of E6(q) or 3·E6(q) in GL(27, q) . . . . . . . . . . . . . . 119

5.6 Overgroups of 2·E7(q) in GL(56, q) . . . . . . . . . . . . . . . . . . . 121

6 Proof of the main theorem 125

6.1 Stabilisers of subspaces of Lie powers . . . . . . . . . . . . . . . . . . 125

6.2 Inducing exceptional Chevalley groups on P/Φ(P ) . . . . . . . . . . . 130

7 Conclusion 135

A GAP and Magma code 139

A.1 Properties of a 3-group in GAP . . . . . . . . . . . . . . . . . . . . . 139

A.2 Exterior square non-isomorphism in Magma . . . . . . . . . . . . . . 139

A.3 Linear algebraic groups in Magma . . . . . . . . . . . . . . . . . . . . 140

A.4 Submodule structures of Lie powers in Magma . . . . . . . . . . . . . 142

Bibliography 149

Index 157

Chapter 1

Introduction

Using standard representation theory, we can consider any given finite group H

as a subgroup of some general linear group defined over the finite field Fp, where p is a

prime. The linear group H is then a group of automorphisms of the associated vector

space V , which is an elementary abelian p-group. In this thesis, we are interested in

an analogue of representation theory that allows us to represent linear groups on non-

abelian p-groups. Namely, we would like to construct a non-abelian p-group P such

that H is the image of the natural action of Aut(P ) on the automorphism group of

the Frattini quotient P/Φ(P ) of P . We denote this image by A(P ), and we call it the

group induced by Aut(P ) on P/Φ(P ). By Burnside’s Basis Theorem [66, Theorem

11.12], we can identify P/Φ(P ) with a vector space over Fp. Indeed, if P/Φ(P ) ∼= V

and if A(P ) = H, then the natural action of A(P ) on P/Φ(P ) is equivalent to the

action of H on V . This is related to the general problem of determining which groups

describe the symmetries of certain mathematical objects. For example, Frucht’s

Theorem [26] states that each finite group is the full automorphism group of some

finite graph.

Suppose now that d > 1 is an integer. Note that if P is a p-group with A(P ) 6

GL(d, p), then P/Φ(P ) ∼= Fdp. Observe also that if A(P ) is a proper subgroup of

GL(d, p), then P cannot be elementary abelian. In 1978, Bryant and Kovacs [14]

proved the following important theorem.

Theorem 1.0.1. Let H be a subgroup of GL(d, p). Then there exists a p-group P

such that A(P ) = H.

Although the above theorem ensures that there exists a p-group associated with

any given subgroup of GL(d, p), it is not obvious whether or not there exists a

“small” associated p-group, in terms of order and other properties such as exponent,

exponent-p class and nilpotency class. Indeed, the p-group constructed in the proof

of Bryant and Kovacs has an exponent-p class comparable to |GL(d, p)|, and an

order that is not explicit but must be huge. We will see that for each linear group

H considered in this thesis, the “smallest” associated p-group is in fact non-abelian.

Since Z(P ) is a characteristic subgroup of a given p-group P , Aut(P ) also induces

a group on the quotient P/Z(P ). Although this induced group is an abstract group,

without an associated linear action in general, it is known that for each finite group

1

2 Chapter 1. Introduction

G, there exists a p-group P of nilpotency class 2 and exponent p2 such that G is

isomorphic to the group induced on P/Z(P ). This was proved by Heineken and

Liebeck [37] in the case of p > 2, and by Hughes [40] in the case of p = 2. On the

other hand, the following problem is open in general:

Problem 1.0.2. Given a subgroup H of GL(d, p), what are the properties of the

“smallest” p-group P such that A(P ) = H?

Bamberg, Glasby Morgan and Niemeyer [7] addressed this problem for maximal

subgroups of GL(d, p) that lie in a certain subset of the Aschbacher classes. The

Aschbacher classes, denoted C1, . . . , C9, describe the maximal subgroups of GL(d, p)

via Aschbacher’s Theorem [3]. Specifically, Bamberg et al. proved:

Theorem 1.0.3. Suppose that p > 3, and let H be a maximal subgroup of GL(d, p)

that does not contain SL(d, p), and that lies in an Aschbacher class of GL(d, p) other

than C6 and C9. Then there exists a p-group P of exponent p, order at most pd4/2

and nilpotency class 2, 3 or 4, such that A(P ) = H.

Bamberg et al. also determined the minimal nilpotency class of the p-group P

on a case-by-case basis. In particular, this minimal class is at most 3 if d > 2.

Moreover, the conclusion of Theorem 1.0.3 holds with several additional hypotheses

when p = 3, and here P has nilpotency class 2. In each case, the exponent-p class of

P is equal to its nilpotency class. However, Problem 1.0.2 is still open in general in

the case where H is not maximal, or where H lies in one of the Aschbacher classes

excluded by Bamberg et al.

It is well-known that the finite simple groups are the “building blocks” of all

finite groups. Therefore, it is natural to study Problem 1.0.2 in the case where H

is such a simple group. Indeed, doing so may lead to a deeper understanding of the

problem for finite groups in general. In this thesis, we consider the case where the

(finite) simply connected version of an exceptional Chevalley group is a subgroup of

H of relatively small index. Speaking generally, the exceptional Chevalley groups

are the finite simple groups of Lie type denoted by G2(q), F4(q), E6(q), E7(q) and

E8(q) for each prime power q (with q > 2 in the first case).

In fact, we began this investigation in our previous work [25]. Here, we proved

that, for each odd prime p, there exists a p-group P of exponent p, exponent-p class

2 and nilpotency class 2, such that A(P ) is the normaliser of G2(p) in GL(7, p),

which is the non-maximal subgroup Z(GL(7, p))G2(p). However, we were not able

to determine in general the minimal order of such a p-group. We also proved the

existence of p-groups of exponent p and nilpotency class 2 or 3 associated with

specific (not necessarily maximal) C6- and C9-subgroups of particular general linear

groups.

We now summarise the main theorem of our thesis, which only considers the case

where q = p. Note that analogous results for the classical Chevalley groups follow

3

from the work of Bamberg et al. summarised in Theorem 1.0.3. However, their work

does not imply any part of our main theorem.

Theorem 1.0.4. Let G be the simply connected version of an exceptional Chevalley

group defined over Fp, with p > r, where r := 3 if G is of type E6 or E7 and r := 2

otherwise. Additionally, let d be the minimal dimension of a nontrivial irreducible

Fp[G]-module. Then there exists a p-group P of exponent-p class r, nilpotency class

r and exponent p, such that A(P ) is the normaliser of G in GL(d, p). Furthermore, if

G is of type G2 or E8, then there exists a p-group P of exponent-p class 2, nilpotency

class 2 and exponent p2, such that A(P ) = G.

For each p-group P mentioned in this theorem, if Q is a p-group with A(Q) =

A(P ), then, roughly speaking, Q is no smaller than P , in terms of exponent-p

class, exponent and nilpotency class. In each case, we construct a p-group with the

specified properties of P as a particular quotient of an associated universal p-group.

Specifically, we construct p-groups of exponent p via the method of Bamberg et

al. [7, §2, §6], and we construct p-groups of exponent-p class 2 via the method of

Glasby, Palfy and Schneider [29, §1–2]. This latter method is an application of the

theory of p-covering groups described by O’Brien [62]. In the cases where r = 2, the

constructed p-group has the smallest order of all groups satisfying the properties of

P . In particular, this holds for the p-group related to G2(p) investigated in [25].

Table 1.0.1 summarises the properties of each constructed p-group P , including the

structure of A(P ). Here, G denotes an exceptional Chevalley group defined over Fp.

Table 1.0.1: The properties of each constructed p-group P from Theorem 1.0.4.Here, ZGL := Z(GL(d, p)), t := (3, p− 1), and r is equal to the exponent-p class andnilpotency class of P .

G d rExponent

of P|P | A(P )

G2(p) 7 2 p p14 ZGLG2(p)

G2(p) 7 2 p2 p14 G2(p)

F4(3) 25 2 3 377 ZGLF4(3)

F4(p), p > 3 26 2 p p78 ZGLF4(p)

E6(p), p > 3 27 3 p p456 (ZGL(t·E6(p))).t

E7(p), p > 3 56 3 p p2508 (ZGL(2·E7(p))).2

E8(p) 248 2 p p496 ZGLE8(p)

E8(p) 248 2 p2 p496 E8(p)

In order to construct each p-group, and in order to show that it is as “small” as

possible, we determine the stabiliser in GL(d, p) of each submodule of the exterior

square of an arbitrary minimal Fp[G]-module V , i.e., a nontrivial irreducible Fp[G]-

module of minimal dimension. When G is of type E6 or E7 and p > 3, we also

determine the stabiliser of each submodule of the third Lie power of V . In fact,

4 Chapter 1. Introduction

the submodule structures of these modules and the stabilisers of these submodules

are of general interest, with many potential applications, such as those described in

[49]. Indeed, we will see in this thesis that we can use this information to determine

overgroups of G in GL(d, q), and to distinguish between G and its overgroups. This

knowledge about overgroups is itself of general interest, for example in the context

of determining the maximal subgroups of classical groups, a problem which has

been studied by Bray, Holt and Roney-Dougal [13], Kleidman and Liebeck [53] and

Schroder [68].

Because of this general interest, we explore the above representation theory in

the more general case where the exceptional Chevalley group G is defined over

an arbitrary finite field Fq (with q a power of p > r). We achieve this by using

highest weight theory to determine the composition factors of the exterior square

or third Lie power of each minimal Fp[G]-module, where G is the linear algebraic

group associated with G, and by determining part of the overgroup structure of G

in GL(d, q). Liebeck [55, §1] gives analogous information about the exterior square

of each minimal module for almost every classical Chevalley group.

The structure of this thesis is as follows. In Chapter 2, we establish preliminary

definitions and results from group theory and representation theory that we require

for our work. We discuss the main problem investigated in this thesis in more

detail in Chapter 3. In this chapter, we also describe the aforementioned universal

p-groups, and explain when (and how) we can induce a given linear group on the

Frattini quotient of a p-group constructed as a quotient of such a universal group.

Chapter 4 begins with a brief description of the simple groups of Lie type, followed

by an explanation of highest weight theory. Later in this chapter, we use highest

weight theory to determine the composition factors of certain modules for linear

algebraic groups, and we subsequently determine the submodule structures of the

relevant Fq[G]-modules. Next, in Chapter 5, we determine part of the overgroup

structure of each group G in the corresponding general linear group. We also use

the description of G2(q) as the automorphism group of the octonion algebra over Fq(see [73, p. 121]) to more closely examine the action of this group on the largest

maximal submodule of the exterior square of a given minimal Fq[G2(q)]-module. In

fact, we prove related results about the larger family of groups G2(F) defined over

arbitrary fields F of characteristic other than 2. In Chapter 6, we use results from

the previous chapters to determine the stabilisers of the submodules of the relevant

Fq[G]-modules, and we then state and prove the full version of our main theorem.

Finally, we summarise our results in Chapter 7, and discuss possible directions for

future research.

Chapter 2

Preliminaries

Throughout this thesis, we use standard notation and elementary definitions

and results related to groups, fields and vector spaces. These can be found in

introductory textbooks in these areas, for example [4, 42, 64, 65, 66]. All vector

spaces are assumed to be finite-dimensional, except where stated otherwise, and

when we refer to a p-group, we mean a finite p-group. Unless otherwise specified,

G denotes an arbitrary group and F denotes an arbitrary field. Also note that,

in general, we denote the image of an element or set x under a map θ by (x)θ.

Furthermore, if we say that a group H acts on G, then we mean that it acts by

automorphisms, and we write HX to denote the (setwise) stabiliser in H of a subset

X of G. We also write gh to denote the image of an element g ∈ G under the action

of an element h ∈ H. In particular, if x and y are elements of the same group, then

xy := y−1xy. Finally, all results and proofs in this thesis are our own work unless

specified otherwise – this includes proofs of results that are not our own.

2.1 Subgroups and quotient groups

In this chapter of the thesis, we present definitions and results related to group

theory and representation theory that we will use throughout the thesis. We begin

with some standard concepts related to subgroups and quotients of groups.

Lemma 2.1.1 (Dedekind’s Identity [72, Lemma 1.2.3]). Let H, J and K be sub-

groups of G such that H 6 K. Then K ∩HJ = H(J ∩K).

Proposition 2.1.2 ([45, p. 131–132]). Let N be a normal subgroup of G, and let

H be a subgroup of Aut(G) that stabilises N . Then H acts naturally on G/N , with

(Nx)φ := Nxφ for all x ∈ G and all φ ∈ H. In particular, if N is characteristic in

G, then this defines a natural action of Aut(G) on G/N .

Definition 2.1.3 ([45, p. 113]). Let H and K be subgroups of G. The commutator

[H,K] of H and K is the subgroup 〈[h, k] | h ∈ H, k ∈ K〉 of G. Here, the

commutator [x, y] of x and y is equal to x−1y−1xy for each x, y ∈ G.

A very important commutator is the derived subgroup G′ := [G,G] of G, which

is characteristic in G. As an example, the derived subgroup of the dihedral group

5

6 Chapter 2. Preliminaries

D10 of order 10 is the subgroup consisting of rotations. If G′ = G, then we say that

G is perfect . In particular, every non-abelian simple group is perfect.

If x, y, z ∈ G, and if H,K and L are subgroups of G, then we write [x, y, z] :=

[[x, y], z] and [H,K,L] := [[H,K], L]. It is also easy to see that [H,K] = [K,H],

and that if H 6 K, then [H,L] 6 [K,L]. Additionally, observe that elements x and

y of G commute if and only if [x, y] = 1. Therefore, z ∈ Z(G) if and only if [z, y] = 1

(or [y, z] = 1) for all y ∈ G. It follows immediately that H 6 Z(G) if and only if

[H,G] = 1. In particular, G is abelian if and only if G′ = 1. We use some of these

observations in the proof of the following elementary proposition.

Proposition 2.1.4. Let z, z′ ∈ Z(G), and let g, g′ ∈ G. Then [gz, g′z′] = [g, g′].

Additionally, if X and Y are subgroups of Z(G), and if H and K are subgroups of

G, then [HX,KY ] = [H,K].

Proof. Using well-known commutator identities [45, p. 114], we have

[gz, g′z′] = [g, g′z′]z[z, g′z′] = [g, g′z′] = [g, z′][g, g′]z′= [g, g′].

Note that HX and KY are groups, since X commutes with H and Y commutes

with K. We have

[HX,KY ] = 〈[gz, g′z] | g ∈ H, g′ ∈ K, z ∈ X, z′ ∈ Y 〉

= 〈[g, g′] | g ∈ H, g′ ∈ K〉 = [H,K].

The next proposition of this section, which shows that homomorphic images of

commutators and commutator groups are equivalent to commutators and commu-

tator groups of homomorphic images, respectively, is well-known.

Proposition 2.1.5. Let H be a group, and let θ : G → H be a homomorphism.

Then for elements x, y ∈ G, ([x, y])θ = [(x)θ, (y)θ]. Furthermore, for subgroups J

and K of G, ([J,K])θ = [(J)θ, (K)θ].

Proof. Since θ is a homomorphism, we have

([x, y])θ = (x−1y−1xy)θ = ((x)θ)−1((y)θ)−1(x)θ(y)θ = [(x)θ, (y)θ].

Thus ([J,K])θ, which is generated by the elements ([j, k])θ for j ∈ J and k ∈ K, is

equal to the group generated by the elements [(j)θ, (k)θ], i.e., [(J)θ, (K)θ].

If N is a normal subgroup of G, then we obtain the following by setting H = G/N

in Proposition 2.1.5, with θ the canonical projection from G to G/N .

Corollary 2.1.6. Let N be a normal subgroup of G, and let J and K be subgroups

of G. Then [NJ/N,NK/N ] = N [J,K]/N .

Corollary 2.1.7 ([45, p. 80]). Let N be a normal subgroup of G. Then G/N is

abelian if and only if G′ 6 N .

2.1. Subgroups and quotient groups 7

Proof. We have [G/N,G/N ] = [NG/N,NG/N ], which is equal to NG′/N by

Corollary 2.1.6. Thus G/N is abelian if and only if NG′/N = 1, which is the

case if and only if G′ 6 N .

We recall that the derived series G(i) of G is the series of subgroups defined by

G(0) := G and G(i+1) := [G(i), G(i)] for each nonnegative integer i. If G is finite, then

we write G∞ to denote the first perfect group in the derived series of G. In other

words, G∞ = G(j), where j is the smallest nonnegative integer such that G(i) = G(j)

for all i > j. Since each group in the derived series of G is characteristic, so is G∞.

It is easy to see that if a finite group G contains a perfect subgroup H, then

H = H ′ = [H,H] 6 [G,G] = G′,

and it follows that H 6 G∞. Moreover, it is well-known that if N is a normal

subgroup of a finite group G such that G/N is soluble, then G∞ = N∞.

Lemma 2.1.8 ([45, Lemma 3.10]). Let H be a group, let θ : G→ H be a homomor-

phism, and let K and N be subgroups of G, with N C6 G. Then ((K)θ)(i) = (K(i))θ

for each nonnegative integer i, and (NK/N)(i) = NK(i)/N . If G and H are finite,

then ((K)θ)∞ = (K∞)θ, and (NK/N)∞ = NK∞/N .

Proof. We prove the lemma by induction on i. When i = 0, ((K)θ)(i) = (K)θ =

(K(i))θ. For the inductive step, we assume that ((K)θ)(i) = (K(i))θ for some i > 0.

Using the inductive hypothesis and Proposition 2.1.5, we have

((K)θ)(i+1) = [((K)θ)(i), ((K)θ)(i)] = [(K(i))θ, (K(i))θ]

= ([K(i), K(i)])θ = (K(i+1))θ.

Therefore, by induction, ((K)θ)(i) = (K(i))θ for each nonnegative integer i. If θ is

the canonical projection from G → G/N , with N a normal subgroup of G, then

this implies that (NK/N)(i) = NK(i)/N . Finally, if G and H are finite, then we

can choose i large enough so that ((K)θ)(i) = ((K)θ)∞ and K(i) = K∞, and the

remaining results follow.

Definition 2.1.9 ([45, p. 117]). The exponent of a group G is the smallest positive

integer k such that xk = 1 for all x ∈ G, when such an integer exists.

Equivalently, the exponent of a group is the least common multiple of the orders

of the group’s elements. Observe that only the trivial group has an exponent of 1.

For a nontrivial example, each element of D10 has order 1, 2 or 5 and hence this

group has exponent 10. Note also that if G has exponent k and if N is a normal

subgroup of G, then (nx)k = 1 for all n ∈ N and x ∈ G. Hence the exponent of

G/N divides the exponent of G.

Khukhro [51, p. 5] states the following definition and related results.


Definition 2.1.10. Let k be a positive integer. Then we define Gk := 〈xk | x ∈ G〉.

Note that this subgroup of G is often denoted Gk, but we avoid this notation so

that this subgroup is not confused with the direct product of k copies of G.

Proposition 2.1.11. For each positive integer k, Gk is a characteristic subgroup of

G.

Proof. Let α ∈ Aut(G). Then, using the properties of automorphisms,

(Gk)α = 〈xk | x ∈ G〉α = 〈(xk)α | x ∈ G〉

= 〈(xα)k | x ∈ G〉 = 〈xk | x ∈ G〉 = Gk.

Thus Gk is characteristic in G.

Lemma 2.1.12. Let N be a proper normal subgroup of G, and let p be a prime.

Then G/N has exponent p if and only if Gp 6 N .

Proof. First, suppose that Gp 6 N . Then Definition 2.1.10 implies that xp ∈ N for

each x ∈ G, and thus (Nx)p = Nxp = N = 1G/N . Hence the order of each element

Nx ∈ G/N divides p. Since G/N is nontrivial, it follows from Definition 2.1.9 that

G/N has exponent p.

Next, suppose that G/N has exponent p. Then for each x ∈ G, we have N(xp) =

(Nx)p = 1G/N = N , and thus xp ∈ N . Thus Gp = 〈xp | x ∈ G〉 6 N .

Definition 2.1.13 ([66, p. 266]). The Frattini subgroup of a finite group G, denoted

by Φ(G), is the intersection of all maximal subgroups of G. If G has no maximal

subgroups, i.e., if G = 1, then we define Φ(G) := 1.

The Frattini subgroup of G is indeed a subgroup of G, as it is an intersection of

subgroups. In fact, Φ(G) is characteristic in G [66, p. 266]. For example, Φ(D10) = 1,

since the subgroup of D10 consisting of rotations is maximal, as is the subgroup

of order 2 generated by a reflection, and these two subgroups intersect trivially.

However, Φ(D8) consists of the identity and the rotation of order 2. We call G/Φ(G)

the Frattini quotient of G [7].

Definition 2.1.14 ([9, Definition 2.3.15]). The rank of a finitely-generated group

is the minimum size of a generating set for the group.

We define the rank of the trivial group to be 0. For a nontrivial example, the

dihedral group D10 can be generated by a rotation and a reflection. As this group

is not cyclic, it has rank 2. Additionally, a d-dimensional vector space has rank d

when considered as an abelian group under vector addition.

We recall that for each positive integer d, the free group F of rank d is the unique

(up to isomorphism) group of rank d such that every group of rank d is a quotient of

F . The following definition describes an important quotient of F that is the “free”

group with respect to certain group theoretic properties.

2.2. Almost simple and quasisimple groups 9

Definition 2.1.15 ([46, p. 242]). Let d and k be positive integers, and let F be

the free group of rank d. The free Burnside group of rank d and exponent k is

B(d, k) := F/F k.

As implied by its name, B(d, k) has rank d and exponent k.

Theorem 2.1.16 ([69, p. 550]). Let d and k be positive integers. Then every group

of rank d and exponent k is a quotient of B(d, k).

Thus B(d, k) can be considered as the “free” group of rank d and exponent k.

This group may be finite or infinite depending on the pair (d, k), and one formulation

of the famous, unsolved Burnside problem asks when B(d, k) is finite [1, p. 806].

Zelmanov [74, 75] won the Fields Medal in 1994 for solving the related Restricted

Burnside Problem and proving that, for each pair (d, k), there exists a finite group

B0(d, k) such that every finite group of rank d and exponent k is a quotient of

B0(d, k) [24].

2.2 Almost simple and quasisimple groups

We now present two generalisations of non-abelian simple groups, as well as a

few related results. Recall that the inner automorphism group Inn(H) of a group H

is a normal subgroup of Aut(H), and Inn(H) ∼= H/Z(H). If H is non-abelian and

simple, then Z(H) = 1, and hence we can identify H = H/Z(H) with Inn(H).

Definition 2.2.1 ([16, p. 16]). A group G is almost simple if there exists a non-

abelian simple group H satisfying H C6 G 6 Aut(H), where we identify H with

Inn(H).

It is clear that each non-abelian simple group is almost simple.

Proposition 2.2.2 ([13, p. 8]). Let H be a finite non-abelian simple group, and let

G be an almost simple group such that H C6 G 6 Aut(H). Then G∞ = H.

The above proposition is a consequence of the Schreier Conjecture, which states

that (Out(H))∞ = 1 for each finite non-abelian simple group H, and whose known

proof relies on the classification of finite simple groups.

Definition 2.2.3 ([13, p. 1]). A group G is quasisimple if G is perfect and G/Z(G)

is non-abelian and simple.

In particular, each non-abelian simple group is quasisimple. The following the-

orem, which applies to finite quasisimple groups, is a seminal result of Schur.

Theorem 2.2.4 ([73, p. 27–28]). Let G be a finite perfect group. Then there exists

a unique (up to isomorphism) largest group J , called the universal cover of G, such

that Z(J) 6 J ′ and J/Z(J) ∼= G.


The centre of J is called the Schur multiplier of G.

Lemma 2.2.5 ([71, Lemma 6.9.2]). Let G be a finite perfect group. Then the

universal cover of G is perfect.

Proof. Let J be the universal cover of G. Then J/Z(J) ∼= G is perfect. Hence

J/Z(J) = (J/Z(J))′, which is equal to Z(J)J ′/Z(J) by Lemma 2.1.8. This in turn

is equal to J ′/Z(J), since Z(J) 6 J ′. It follows that J = J ′, as required.

It follows immediately that if G is a finite non-abelian simple group, then the

universal cover of G is quasisimple.

For the following lemma, recall that the centre of a group J is a characteristic

subgroup, and hence Aut(J) acts naturally on J/Z(J) by Proposition 2.1.2.

Lemma 2.2.6 ([32, Corollary 5.1.4]). Let J be the universal cover of a finite qua-

sisimple group G. Then each automorphism of G ∼= J/Z(J) lifts to a unique au-

tomorphism of J that stabilises Z(J). Specifically, for each α ∈ Aut(J/Z(J)) ∼=Aut(G), there exists a unique automorphism α of J such that Z(J)xα = (Z(J)x)α

for all x ∈ J . If G is simple, then the natural action θ : Aut(J) → Aut(J/Z(J))

is an isomorphism, i.e., any given automorphism of J is equal to α for a unique

automorphism α of J/Z(J) ∼= G.

Observe that if G is simple, then (Inn(J))θ = Inn(J/Z(J)), and hence

Out(J) = Aut(J)/Inn(J) ∼= Aut(J/Z(J))/Inn(J/Z(J)) = Out(J/Z(J)).

2.3 Nilpotent groups

In this section, we introduce nilpotent groups. Our first definition is stated by

Gorenstein [30, p. 21], but we use the notation of Leedham-Green and McKay [54,

p. 6], which is more standard.

Definition 2.3.1. The lower central series of G is γi(G) | i = 1, 2, . . ., where γ1 :=

G and γi+1(G) := [γi(G), G] for each i > 1. If γi+1(G) = 1 for some nonnegative

integer i, then we say that G is nilpotent , and the smallest such i is called the

nilpotency class of G.

Note that for each positive integer i, γi(G) is a characteristic subgroup of G, and

γi+1(G) 6 γi(G) [30, Theorem 3.1].

There are several alternative equivalent definitions of a nilpotent group and the

nilpotency class of such a group – for example, see [45, Ch. 1D]. Observe that

G is nilpotent of class 0 if and only if G = 1. We also have γ2(G) = G′, and

it follows that G is abelian if and only if it is nilpotent of class at most 1. More

generally, a nontrivial group with a small nilpotency class is “closer” to being abelian

than one with a large nilpotency class, and hence nilpotency class is a measure of

2.4. p-groups 11

“abelianness”. An example of a non-abelian nilpotent group is the dihedral group

D16, which has nilpotency class 3 [54, p. 53]. It also follows from the definition of

the derived series of a group that each nilpotent group is soluble. However, there

exist groups that are soluble but not nilpotent, e.g., the symmetric group S3.

Lemma 2.3.2 ([12, p. 71–72]). Let i be a positive integer, let H be a group, and let

θ : G→ H be an epimorphism. Then γi(H) = (γi(G))θ. In particular, if H = G/N

for a normal subgroup N of G, then γi(G/N) = Nγi(G)/N . Moreover, if G is a

nilpotent group of class r, then each homomorphic image of G and each quotient of

G is nilpotent of class at most r.

The following result generalises Corollary 2.1.7.

Proposition 2.3.3. Let N be a normal subgroup of G, and let i > 1 be an integer.

Then G/N is nilpotent of class i if and only if N contains γi+1(G) but not γi(G).

Proof. We have from Definition 2.3.1 that G/N is nilpotent of class i if and only

if γi+1(G/N) is trivial and γi(G/N) is nontrivial. Lemma 2.3.2 shows that this first

condition is equivalent to 1 = Nγi+1(G)/N , i.e., Nγi+1(G) = N , i.e., γi+1(G) 6 N .

The second condition is equivalent to 1 < Nγi(G)/N , i.e., N < Nγi(G), using the

Correspondence Theorem. This in turn is equivalent to γi(G) 66 N .

2.4 p-groups

We recall that, for a prime p, a p-group is a finite group of order pa for some

nonnegative integer a. In this section, we explore some properties of p-groups.

Lemma 2.4.1 ([45, p. 21]). Every p-group is nilpotent.

This means that each p-group has a well-defined nilpotency class.

Definition 2.4.2 ([45, p. 27]). A p-group is elementary abelian if it is abelian and

if every nontrivial element has order p.

For example, every nonidentity element in the abelian 5-group Z5 has order

5, and hence this group is elementary abelian. In fact, it follows from the Fun-

damental Theorem of Finite Abelian Groups that, for a fixed prime p, the set of

(finite) elementary abelian p-groups (up to isomorphism) is equal to the set of finite

(nonempty) direct products of copies of Zp. Note also that a nontrivial abelian

p-group is elementary abelian if and only if it has exponent p.

Theorem 2.4.3 ([45, Lemma 4.5]). Let N be a normal subgroup of a p-group P .

Then P/N is elementary abelian if and only if Φ(P ) 6 N .

Proposition 2.4.4 ([51, p. 60]). Let P be a p-group. Then Φ(P ) = P ′P p.


Proof. Since P ′ and P p are normal subgroups of P , so is P ′P p. Furthermore,

Corollary 2.1.7 and Lemma 2.1.12 imply that P/(P ′P p) is the largest elementary

abelian quotient of P . Thus Φ(P ) = P ′P p by Theorem 2.4.3.

Now, it follows from Definitions 2.1.9 and 2.1.10 that P p = 1 when P is a group

of exponent p. Proposition 2.4.4 therefore yields the following.

Corollary 2.4.5. Let P be a group of exponent p. Then Φ(P ) = P ′.

The next theorem allows us to identify the Frattini quotient of a given p-group

with a vector space over Fp.

Theorem 2.4.6 (Burnside’s Basis Theorem [66, Theorem 11.12]). Let P be a non-

trivial p-group of rank d. Then we can identify P/Φ(P ) with the vector space Fdp. In

particular, |P/Φ(P )| = pd. Furthermore, a subset x1, . . . , xd of P is a generating

set for P if and only if Φ(P )x1, , . . . ,Φ(P )xd is a vector space basis for P/Φ(P ).

It follows from Theorem 2.4.3 that the vector space Fdp is elementary abelian. In

fact, any vector space V over a finite field of characteristic p is elementary abelian,

since kv = 0V for a nonnegative integer k and a vector v ∈ V if and only if p | k or

v = 0V .

Proposition 2.4.7. Let P be a nontrivial p-group, and let N be a normal subgroup

of P . Then P and P/N have the same rank if and only if N 6 Φ(P ).

Proof. By Burnside’s Basis Theorem, it suffices to show that P/Φ(P ) is isomorphic

to (P/N)/Φ(P/N) if and only if N 6 Φ(P ). Suppose that N 6 Φ(P ). Then

Φ(P/N) = Φ(P )/N [43, Hilfssatz III.3.4], and (P/N)/Φ(P/N) = (P/N)/(Φ(P )/N)

is isomorphic to P/Φ(P ) by the Third Isomorphism Theorem.

Conversely, suppose that P/Φ(P ) ∼= (P/N)/Φ(P/N). By the Correspondence

Theorem, Φ(P/N) = K/N for some subgroup K of P that contains N . Hence

NΦ(P )/N 6 K/N [43, Hilfssatz III.3.4], and the Correspondence Theorem gives

NΦ(P ) 6 K. Thus K contains Φ(P ). Additionally, P/Φ(P ) is isomorphic to

(P/N)/(K/N), and by considering group orders, we have |Φ(P )| = |K|. Therefore,

K = Φ(P ). We have shown that NΦ(P ) 6 Φ(P ), and so N 6 Φ(P ).

We conclude this section by discussing an important series of subgroups of a

p-group.

Definition 2.4.8 ([21, p. 2272]). Let P be a p-group. The lower exponent-p central

series of P is the series of groups Pi(P ) for each positive integer i, where P1(P ) := P

and Pi+1(P ) := [Pi(P ), P ]Pi(P )p for each i > 1. The smallest nonnegative integer i

such that Pi+1(P ) = 1 is the exponent-p class of P .

2.4. p-groups 13

Note that Pi+1(P ) 6 Pi(P ) for all i, that Pi(P ) is a characteristic subgroup of

P , and that there does indeed exist an integer i such that Pi(P ) = 1 [39, p. 355–

356]. In particular, a nontrivial p-group has exponent-p class 1 if and only if it is

elementary abelian.

The first result of the following proposition is stated by O’Brien [62, p. 678],

while the remaining three results are our own.

Proposition 2.4.9. Let P be a p-group of exponent-p class c. Then:

(i) P has nilpotency class at most c;

(ii) P has exponent at most pc;

(iii) if P is abelian, then P has exponent pc; and

(iv) if P has exponent p, then P has nilpotency class c.

Proof. Let µi(P ) be the series of subgroups of P defined by µ1(P ) := P and

µi+1(P ) := (µi(P ))p for each positive integer i. Note that γ2(P ) = [P, P ] =

[P1(P ), P ], and that µ2(P ) = P p = P1(P )p. By induction on i, we will prove

that γi+1(P ) 6 [Pi(P ), P ] and that µi+1(P ) 6 (Pi(P ))p for each i. For the inductive

step, suppose that these containments hold for a particular positive integer i. Then

each of γi+1(P ) and µi+1(P ) is a subgroup of [Pi(P ), P ]Pi(P )p, which is equal to

Pi+1(P ) by Definition 2.4.8. We therefore have

γi+2(P ) = [γi+1(P ), P ] 6 [Pi+1(P ), P ]

and

µi+2(P ) = (µi+1(P ))p 6 (Pi+1(P ))p.

This completes our proof by induction. In particular, γc+1(P ) 6 [Pc(P ), P ] 6

Pc+1(P ) and µc+1(P ) 6 (Pc(P ))p 6 Pc+1(P ). As P has exponent-p class c, we

have 1 = Pc+1(P ) = γc+1(P ) = µc+1(P ). Thus P has nilpotency class at most c.

Furthermore, observe that P pc lies in µc+1(P ). Hence P pc = 1, which means that P

has exponent at most pc.

Suppose now that P is abelian, and let x1, . . . , xm be a generating set for

some subgroup H of P , where m is a positive integer. Then as P is abelian, we have

H pi = 〈xpi

1 , . . . , xpi

m〉 for each positive integer i. It follows that (P pi)p = P pi+1for

each i. Since µ1(P ) = P , a simple inductive argument shows that µi+1(P ) = P pi for

each i. Furthermore, since P is abelian, we have [H,P ] = 1 for each subgroup H of

P , and hence Pi(P ) = µi(P ) for each i. Therefore, P pi = Pi+1(P ) for each i, and so

P has exponent pc. If we instead suppose that P has exponent p, then H p = 1 for

any subgroup H of P . It follows that Pi(P ) = γi(P ) for each positive integer i, and

thus the nilpotency class of P is c.


Note that it is possible for a p-group P with exponent-p class c to have nilpotency

class less than c and exponent less than pc. For example, calculations in the GAP

[28] computer algebra system using the code in Appendix A.1 show that the 3-group

with presentation

〈a, b | a9 = b9 = [a, b][a, b−1] = [a, b][a−1, b] = [a3, b3] = 1〉

and order 36 = 729 has exponent-p class 3, nilpotency class 2, and exponent 32.

2.5 Forms

In this section, we discuss bilinear, σ-Hermitian and quadratic forms, which

we will use in Section 2.6 to define some of the classical groups. All definitions

and results here are from Bray, Holt and Roney-Dougal [13, Ch. 1.5–1.6, Ch. 1.8],

except where stated otherwise. Throughout this section, V is the d-dimensional

vector space over the field F, where d is a positive integer.

Definition 2.5.1. A bilinear form on V is an F-bilinear map β : V ×V → F. We say

that β is alternating (or symplectic) if (v, v)β = 0 for all v ∈ V . If (u, v)β = (v, u)β

for all u, v ∈ V , then we call β symmetric (or orthogonal if char(F) > 2). Finally,

β is reflexive if, for all u, v ∈ V , (u, v)β and (v, u)β are either both zero or both

nonzero.

Theorem 2.5.2. Each reflexive bilinear form on V is symmetric or alternating.

Definition 2.5.3. Suppose that F has an automorphism σ of order 2. Then a σ-

Hermitian form (or unitary form) on V is a map β : V × V → F such that, for all

u, v, w ∈ V and all λ, µ ∈ F:

(i) (u, v + w)β = (u, v)β + (u,w)β;

(ii) (u+ v, w)β = (u,w)β + (v, w)β;

(iii) (λu, µv)β = λµσ(u, v)β; and

(iv) (v, u)β = ((u, v)β)σ.

In particular, if F = Fq, then such a form is only defined when q1/2 is a power of

a prime, and in this case σ is the automorphism of F that maps each scalar µ ∈ Fto µq

1/2.

Note that the zero form, which maps (u, v) to 0 for all u, v ∈ V , is a bilinear

form and a σ-Hermitian form (when the automorphism σ exists).

Definition 2.5.4. A quadratic form on V is a map Q : V → F such that:

(i) (λv)Q = λ2(v)Q for all λ ∈ F and v ∈ V ; and

(ii) the map β, defined by (u, v)β := (u+ v)Q− (u)Q− (v)Q for all u, v ∈ V , is a

symmetric bilinear form.

2.5. Forms 15

We call β the polar form of Q.

For the remainder of this section, when we say “a form”, we mean a bilinear form,

a σ-Hermitian form or a quadratic form. Observe that if a given form is defined on

V , then its restriction to any subspace of V is a form on that subspace.

Definition 2.5.5 ([73, p. 55]). Let β be a bilinear or σ-Hermitian form on V , and

let U be a subspace of V . Then the orthogonal complement of U (with respect to

β) is U⊥ := v ∈ V | (u, v)β = 0 for all u ∈ U. The radical of β is rad(β) := V ⊥.

It is clear from the definitions of bilinear and σ-Hermitian forms that U⊥ is a

subspace of V . Furthermore, if rad(β) = V , then β is the zero form.

Definition 2.5.6. We say that a bilinear or σ-Hermitian form β is non-degenerate

if rad(β) = 0. Otherwise, β is degenerate. A quadratic form is non-degenerate if

its polar form is non-degenerate, and it is degenerate otherwise.

Proposition 2.5.7. Suppose that there exists a non-degenerate alternating bilinear

form on V . Then d = dim(V ) is even.

We recall that the general linear group GL(d,F) is the group of invertible d ×d matrices in F, and that this group can be identified with the general linear

group GL(V ) of invertible linear transformations from V to itself. We also write

GL(d, q) := GL(d,Fq), where q is a power of a prime.

Definition 2.5.8. Let H be a subgroup of GL(d,F). Then H is a group of simi-

larities of a bilinear or σ-Hermitian form β (respectively, a quadratic form Q) on V

if, for each h ∈ H, there exists a scalar λ ∈ F \ 0 such that (uh, vh)β = λ(u, v)β

(respectively, (vh)Q = λ(v)Q) for all u, v ∈ V . Here, we say that H preserves the

form up to scalars . If λ = 1 for all h ∈ H, then we say that H preserves the form

(or preserves the form absolutely), and that H is a group of isometries of the form.

Observe that if a subgroup H of GL(d,F) preserves a form up to scalars or ab-

solutely, then each subgroup of H also preserves all scalar multiples of that form

up to scalars or absolutely, respectively. In each case, if Z is a nontrivial subgroup

of Z(GL(d,F)), then the group ZH preserves the form up to scalars (but not abso-

lutely). This is because Z(GL(d,F)) consists of the nonzero d × d scalar matrices

over F, i.e., the multiples of the d×d identity matrix by nonzero scalars in F. More-

over, if a subgroup of GL(d,F) preserves a quadratic form up to scalars or absolutely,

then it also preserves its polar form up to scalars or absolutely, respectively.

The following result is well-known.

Proposition 2.5.9. Let β be a bilinear or σ-Hermitian form on V , and let H be a

subgroup of GL(d,F) that preserves β and stabilises a subspace U of V . Then H

also stabilises U⊥.


Proof. Let h ∈ H, and let v ∈ U⊥. Then h−1 ∈ H, and for each u ∈ U , we have

uh−1 ∈ U and hence (uh

−1, v)β = 0 by Definition 2.5.5. Definition 2.5.8 implies that

0 = ((uh−1

)h, vh)β = (u, vh)β for each u ∈ U , and hence vh ∈ U⊥. Therefore, H

stabilises U⊥.

Definition 2.5.10. Let e1, e2, . . . , ed be a basis for V . Then the matrix of a

bilinear or σ-Hermitian form β (with respect to the chosen basis) on V is the matrix

Jβ := (bij)di,j=1, where bij := (ei, ej)β for all i, j ∈ 1, 2, . . . , d.

Proposition 2.5.11. Let β be a bilinear or σ-Hermitian form on V . Then β is

non-degenerate if and only if det(Jβ) (with respect to any choice of basis for V ) is

nonzero.

Definition 2.5.12. Suppose that d is even, and that F is a finite field. In addition,

let β be a non-degenerate orthogonal form on V , and let A be the d-dimensional

anti-diagonal matrix with each anti-diagonal entry equal to 1. Then β has plus type

if det(Jβ) (with respect to any choice of basis for V ) and det(A) are both squares

in F or both not squares in F. Otherwise, β has minus type.

If F = Fq and if d is odd, then we will say that a non-degenerate orthogonal form

on V has type .We now state another well-known result.

Proposition 2.5.13. Let H be a subgroup of GL(d,F) that preserves a bilinear or

σ-Hermitian form β (respectively, a quadratic form Q) on V up to scalars. Addition-

ally, let x ∈ GL(d,F), and define the conjugate form βx−1 : V ×V → F (respectively,

Qx−1 : V → F) to be the map defined by (u, v)βx−1 := (ux−1, vx

−1)β (respectively,

(v)Qx−1 := (vx−1

)Q) for all u, v ∈ V . Then the conjugate form is a form on V of

the same type as the original form, and the subgroup Hx preserves the conjugate

form up to scalars. If H preserves the original form absolutely, then Hx preserves

the conjugate form absolutely.

Proof. First, the action of x−1 on V preserves addition and scalar multiplication.

Moreover, the image under x−1 of any basis for V is itself a basis. It follows from

Definitions 2.5.1, 2.5.3, 2.5.4 and 2.5.12 and Proposition 2.5.11 that the conjugate

form is a form of the same type as the original form. Now, consider a bilinear or

σ-Hermitian form β on V . Since H preserves β up to scalars, there exists a scalar

λ ∈ F for each h ∈ H such that, for all u, v ∈ V ,

(uhx

, vhx

)βx−1 = (ux−1hx, vx

−1hx)βx−1 = (ux−1h, vx

−1h)β

= ((ux−1

)h, (vx−1

)h)β = λ(ux−1

, vx−1

)β = λ(u, v)βx−1 .

Thus Hx preserves βx−1 up to scalars. If H preserves β absolutely, then λ = 1 for

each h ∈ H, and hence Hx preserves βx−1 absolutely. The proof in the case of a

2.6. Classical groups 17

quadratic form Q on V is similar, except that Q is a function of one vector instead

of two.

We recall now that the special linear group SL(d,F) is the subgroup of GL(d,F)

consisting of all matrices of determinant 1. We also write SL(d, q) := SL(d,Fq).

Proposition 2.5.14. Suppose that d > 2, with F = Fq for q a power of an odd

prime. Then SL(d, q) does not preserve any non-degenerate form up to scalars.

2.6 Classical groups

We now define several of the classical groups , which are associated with vector

spaces, and with forms on those vector spaces. Throughout this section we consider

the d-dimensional vector space V over the finite field Fq, where d is a positive

integer and q is a prime power. Except where stated otherwise, the definitions in

this section are from Bray, Holt and Roney-Dougal [13, Ch. 1.5–1.6], who provide an

extensive discussion of the classical groups, including groups that we do not define

here. However, we use notation consistent with Robinson [64].

First, the groups GL(d, q) and SL(d, q) are classical groups associated with the

zero form on V . We recall that, for a general field F, the centre of GL(d,F) consists of

the nonzero d×d scalar matrices over F, while Z(SL(d,F)) = Z(GL(d,F))∩SL(d,F).

It is clear that |Z(GL(d, q))| = q − 1. The following proposition gives the order of

|Z(SL(d, q))|. Here, and throughout this thesis, we write (x, y) := gcd(x, y) for

integers x and y.

Proposition 2.6.1 ([73, p. 78]). The centre of SL(d, q) has order (q − 1, d).

Proposition 2.6.2 ([31, p. 7]). Suppose that (d, q) 6= (2, 2). Then SL(d, q) is the

derived subgroup of GL(d, q).

Definition 2.6.3. Suppose that d is even. Then the symplectic group Sp(d, q) is

the unique (up to isomorphism) largest group of isometries in GL(d, q) of a non-

degenerate alternating form on V .

More precisely, each non-degenerate alternating form on V is preserved by a

unique copy of Sp(d, q) in GL(d, q). The requirement that d is even is a consequence

of Proposition 2.5.7, and we will implicitly assume that this requirement is met

whenever we refer to the group Sp(d, q). We also have Sp(d, q) 6 SL(d, q) [13,

Theorem 1.6.7].

The following proposition follows from the data in [13, Table 1.3] and from [73,

p. 64, p. 78].

Proposition 2.6.4. The centre of Sp(d, q) is equal to Sp(d, q) ∩ Z(GL(d, q)), and

has order (q − 1, 2).


Definition 2.6.5. Suppose that q is a square. The general unitary group GU(d, q1/2)

is the unique (up to isomorphism) largest group of isometries in GL(d, q) of a

non-degenerate σ-Hermitian form on V . Additionally, the special unitary group

SU(d, q1/2) is defined to be SL(d, q) ∩GU(d, q1/2).

Although the general and special unitary groups are subgroups of GL(d, q), it is

standard to denote these subgroups using q1/2 and not q. Equivalently, the group

SU(d, q) is a subgroup of GL(d, q2).

Definition 2.6.6. Suppose that q is odd and that d > 1. If d is odd, then the general

orthogonal group, denoted by GO(d, q) or GO(d, q), is the unique (up to isomor-

phism) largest group of isometries in GL(d, q) of a non-degenerate orthogonal form

on V . If d is even, then the general orthogonal groups are GO+(d, q) and GO−(d, q),

with each the unique (up to isomorphism) largest group of isometries in GL(d, q)

of a non-degenerate orthogonal form on V of plus or minus type, respectively. For

ε ∈ ,+,−, the special orthogonal group SOε(d, q) is equal to SL(d, q)∩GOε(d, q).

Definition 2.6.7. Suppose that q is odd, that d > 1, and that ε ∈ ,+,−.The generally quasisimple classical group Ωε(d, q) is the unique (normal, maximal)

subgroup of SOε(d, q) of index 2. Specifically, this subgroup is the kernel of the

spinor norm, which is a homomorphism from SOε(d, q) to the group of order 2.

Although we do not provide one here, an explicit definition of the spinor norm

is given Bray, Holt and Roney-Dougal [13, p. 28]. The groups GO±(d, q), SO±(d, q)

and Ω±(d, q) can also be defined when d and q are both even. In this case, these

groups preserve a non-degenerate quadratic form on V . Note that we may write

SO(d, q) = SO(d, q) and Ω(d, q) = Ω(d, q). We will implicitly assume that q is odd

whenever we refer to the groups GO(d, q), SO(d, q) and Ω(d, q).

The name of the group Ωε(d, q) is related to the following proposition.

Proposition 2.6.8 ([13, Ch. 1.10]). Let S ∈ SL(d, q), SU(d, q), Sp(d, q),Ωε(d, q),with d > 1 and ε ∈ ,+,−. Additionally, suppose that (d, q) /∈ (2, 2), (2, 3),and that S /∈ SU(3, 2), Sp(4, 2),Ω±(2, q),Ω(3, 3),Ω+(4, q). Then S is quasisimple.

In fact, the following result, primarily stated by Wilson [73, p. 80], shows that

Ωε(d, q) is often a non-abelian simple group. Note that although Wilson does not

state this in the context of this result, the group Ω(3, 3) is not simple [53, Proposition

2.9.2]. While Wilson states the order of the centre of Ω±(d, q), the final fact in this

proposition follows from the data in [13, Table 1.3].

Proposition 2.6.9. Suppose that q is odd. If d > 1 is odd, with (d, q) 6= (3, 3),

then Ω(d, q) is non-abelian and simple. If d is even, then Ω+(d, q) is non-abelian

and simple if and only if qd/2 ≡ 3 (mod 4), and Ω−(d, q) is non-abelian and simple

if and only if qd/2 ≡ 1 (mod 4). If Ω±(d, q) is not simple, then its centre, which is a

proper subgroup of order 2, is equal to Ω±(d, q) ∩ Z(GL(d, q)).

2.7. Tensor powers and exterior powers of vector spaces 19

We now state a well-known result.

Proposition 2.6.10. Let ε ∈ ,+,−, and suppose that d > 2, with (d, q) 6= (3, 3)

and (d, ε) 6= (4,+). Then Ωε(d, q) = (SOε(d, q))′ = (GOε(d, q))∞.

Proof. Let G := GOε(d, q), let S := SOε(d, q), and let Ω := Ωε(d, q). As Ω is

perfect by Proposition 2.6.8, we have Ω = Ω′ 6 S ′. On the other hand, since S/Ω

is abelian of order 2 by Definition 2.6.7, Corollary 2.1.7 implies that S ′ 6 Ω. Hence

S ′ = Ω. Next, |G : S| 6 2 [13, Table 1.3], and hence S C6 G, with G/S abelian of

order at most 2. This means that G/S is soluble, and so G∞ = S∞ = Ω∞ = Ω.

In fact, in most cases, Ωε(d, q) is the derived subgroup of GOε(d, q) [15, p. 1247–

1248].

Definition 2.6.11 ([16, Ch. 2]). Let G := GL(d, q), let ZGL := Z(G), and let H

be a subgroup of G. Then the projective group H corresponding to H is the image

of H under the canonical projection from G to G/ZGL, i.e., H = ZGLH/ZGL. For a

classical subgroup of G denoted by X(d, q), we often write PX(d, q) := X(d, q).

It follows from the Second Isomorphism Theorem that H ∼= H/(H∩ZGL). Exam-

ples of projective groups are the projective general linear group PGL(d, q) = GL(d, q)

and the projective special linear group PSL(d, q) = SL(d, q). The Correspondence

Theorem implies that H is a subgroup of PGL(d, q). Note that if H = SU(d, q) 6

GL(d, q2), then H = PSU(d, q) 6 PGL(d, q2).

2.7 Tensor powers and exterior powers of vector spaces

Here, we discuss tensor powers and exterior powers of vector spaces. Throughout

this section, we use terminology and notation consistent with Bamberg et al. [7].

We assume that the reader is familiar with standard definitions and results related

to tensor products of vector spaces, such as those that can be found in [65, Ch. 14].

In particular, if U and V are vector spaces over the same field with bases R and S,

respectively, then r⊗ s | r ∈ R, s ∈ S is a basis for the tensor product U ⊗V of U

and V , where the tensor r⊗ s is the tensor product of the vectors r ∈ R and s ∈ S.

Thus dim(U ⊗ V ) = dim(U) dim(V ). We also define tensor products of multiple

vector spaces V1, . . . , Vk inductively by V1 ⊗ · · · ⊗ Vj := (V1 ⊗ · · · ⊗ Vj−1) ⊗ Vj

for each j ∈ 3, 4, . . . , k. Using the same method, we define v1 ⊗ · · · ⊗ vk with

vi ∈ Vi for each i, and we extend this definition multilinearly to define u⊗ vk, where

u ∈ (V1 ⊗ · · · ⊗ Vk−1).

The following definition is from Greub [35, p. 61]. Although Greub works over

fields of characteristic 0, all definitions and results in this section apply over all

fields, except where stated otherwise.


Definition 2.7.1. Let k be a positive integer, and let V is a vector space over F.

The k-th tensor power , denoted by T kV , is the tensor product of k copies of V . We

also write T 0V := F.

It is clear that for all nonnegative integers j and k, T j+kV = T jV ⊗ T kV , and

dim(T kV ) = dim(V )k.

Definition 2.7.2 ([35, Ch. 4.1, Ch. 5.3]). Let V be a vector space, and let k > 2

be an integer. Furthermore, let NkV be the subspace of T kV spanned by the set

of all tensor products v1 ⊗ · · · ⊗ vk such that v1, . . . , vk ∈ V and such that there

exist integers i, j ∈ 1, . . . , k with i 6= j and vi = vj. Then the k-th exterior power

of V is the quotient space AkV := (T kV )/(NkV ). We also define an associated

multilinear map Λk : V × · · · × V︸︷︷︸k times

→ AkV by

(v1, v2, . . . , vk)Λk := v1 ∧ v2 ∧ · · · ∧ vk := (v1 ⊗ v2 ⊗ · · · ⊗ vk) +NkV

for v1, . . . , vk ∈ V .

We call v1∧v2∧· · ·∧vk the wedge product of the vectors v1, v2, . . . , vk [65, p. 393].

The exterior power A2V is also called the exterior square of V [22, p. 12].

Lemma 2.7.3 ([35, p. 105]). Let V be a vector space of positive dimension d

with basis e1, . . . , ed, and let k > 2 be an integer. Then a basis for AkV is

ei1 ∧ · · · ∧ eik | 1 6 i1 < · · · < ik 6 d, and dim(AkV ) =(dk

).

Proposition 2.7.4 ([12, p. 511]). Let U and W be vector spaces over the same

field, and let k > 2 be an integer. Furthermore, let f : Uk → W be a multilinear

map that is alternating, i.e., satisfying the property that (u1, u2, . . . , uk)f = 0 for

all u1, u2, . . . , uk ∈ U with ui = ui+1 for some i ∈ 1, 2 . . . , k− 1. Then there exists

a unique linear map f : AkU → W such that (u1∧u2∧· · ·∧uk)f = (u1, u2, . . . , uk)f

for all u1, u2, . . . , uk ∈ U .

Proposition 2.7.5 ([7, p. 2937]). Let k > 2 be an integer, and let V be a vector

space over a field whose characteristic is equal to 0 or greater than k. Then we can

identify AkV with a subspace of T kV .

Definition 2.7.6 ([48, p. 225]). An algebra over F is a vector space U over Fequipped with a binary multiplication such that, for all x, y, z ∈ U and all a ∈ F:

(i) (x+ y)z = xz + yz and x(y + z) = xy + xz; and

(ii) a(xy) = (ax)y = x(ay).

Note that the multiplication in the above definition is not necessarily associative.

For a basic example of an algebra, let d be a positive integer. If we consider the

set of d × d matrices over a field F as the vector space Fd2 , then this vector space

2.8. Representation theory 21

is an associative algebra when equipped with standard matrix multiplication. The

following definition gives another example of an associative algebra.

Definition 2.7.7 ([35, p. 62]). Let V be a vector space. The infinite-dimensional

tensor algebra over V is T (V ) :=⊕

i>0 TiV . The elements of T (V ) are

(v0, v1, . . .) | vi ∈ T iV for each i > 0, vi 6= 0 for only finitely many values of i.

Additionally, T (V ) is equipped with entrywise addition and the bilinear multipli-

cation (u0, u1, . . .)(v0, v1, . . .) := (w0, w1, . . .), where for each integer k > 0, wk :=∑i,j>0i+j=k

ui ⊗ vj.

For each nonnegative integer k, we can identify T kV with the subspace of T (V )

given by (v0, v1, . . .) ∈ T (V ) | vi = 0 for all i 6= k. In particular, we identify

each tensor u ∈ T kV with the element (v0, v1, . . .) ∈ T (V ) such that vk = u and

vi = 0 for all i 6= k. Now, let j and k be nonnegative integers, let u ∈ T jV ,

and let v ∈ T kV . Then with u and v considered as elements of T (V ), we have

uv = (w0, w1, . . .), with wj+k = uj ⊗ vk and with wi = 0 for all i 6= j + k. Thus the

element uv ∈ T (V ) corresponds via our identification to u ⊗ v ∈ T j+kV . There is

therefore a natural identification between tensor products of tensors in tensor powers

of V and products of elements in the corresponding subspaces of T (V ). It follows

that the above definitions and results related to tensor powers of V hold when these

tensor powers are considered as subspaces of T (V ). For example, if k > 2 is an

integer, and if F is a field whose characteristic is either 0 or greater than k, then we

can identify AkV with a subspace of T (V ).

2.8 Representation theory

We now discuss concepts from representation theory, which is used to represent

a given group as a subgroup of a general linear group. We assume that the reader is

familiar with standard definitions and results from representation theory, for exam-

ple from [44, Ch. 1-2] and [65, Ch. 4]. Whenever we refer to a module in this thesis,

we mean an F[G]-module.

Note that if ρ : G → GL(V ) is a representation of G, then we say that the

dimension of ρ is dim(V ), as per the terminology of Bray, Holt and Roney-Dougal

[13, p. 37]. However, many authors refer to dim(V ) as the degree of ρ. It is also

important to note that each F[G]-module is a vector space V associated with a linear

action of G on V , extended linearly in the group algebra F[G]. Hence we will often

write vx to denote the image of (v, x) under the map from V ×F[G] to V associated

with such a module.

We refer to the 0-dimensional F[G]-module as the zero module. In addition,

the trivial irreducible F[G]-module is the unique (up to isomorphism) 1-dimensional


F[G]-module associated with the trivial action of G on V , i.e., with vg = v for all

v ∈ V and all g ∈ G.

Definition 2.8.1 ([13, Definition 1.8.1]). Let d be a positive integer, and let ρ1

and ρ2 be d-dimensional F-representations of G. Then, identifying the codomain of

each of ρ1 and ρ2 with GL(d,F), we say that ρ1 and ρ2 are equivalent if there exists

α ∈ GL(d,F) such that (g)ρ1 = α−1(g)ρ2α for all g ∈ G.

It is well-known that each F[G]-module affords an F-representation of G, and

each F-representation of G is afforded by an F[G]-module. Furthermore, two F[G]-

modules are isomorphic if and only if they afford equivalent F-representations. Also

observe that if H is the image in GL(V ) of the representation afforded by an F[G]-

module V , and if U is a subspace of V , then U is a submodule of V if and only if

H lies in the stabiliser GL(V )U of U in GL(V ).

Our next definition is adapted from Kleidman and Liebeck [53, p. 200].

Definition 2.8.2. A minimal F[G]-module is a nontrivial irreducible F[G]-module

of minimal dimension.

For example, up to isomorphism, there are exactly four nontrivial irreducible

modules for the alternating group A5 over C, of dimension 3, 3, 4 and 5, respec-

tively [6, p. 125]. Thus the minimal C[A5]-modules are the irreducible modules of

dimension 3.

Parker and Rowley [63, p. 40] state the following result applied to bilinear forms.

We will show that it also applies to σ-Hermitian forms.

Proposition 2.8.3. Let V be an irreducible G-module, and suppose that G pre-

serves a degenerate bilinear or σ-Hermitian form β on V . Then β is the zero form.

Proof. The group G acts linearly on the G-module V , and hence Proposition 2.5.9

shows that G stabilises the orthogonal complement V ⊥ of V with respect to β. This

orthogonal complement is therefore a submodule of V , and it is equal to rad(β)

by Definition 2.5.5. As β is non-degenerate, Definition 2.5.6 gives rad(β) 6= 0. It

follows from the irreducibility of V that rad(β) = V , and thus β is the zero form.

Now, if V is the d-dimensional vector space over the field F, then there exists

a faithful d-dimensional F-representation of G if and only if G is isomorphic to a

subgroup of GL(V ). Observe that if ρ : G→ GL(V ) is a representation, and if H is

a normal subgroup of G that lies in ker ρ, then the map ρ′ : G/H → GL(V ) given by

(Hg)ρ′ := (g)ρ for all g ∈ G is another well-defined representation. Also note that

if a G-module V affords a representation ρ, then ker ρ is the set of elements g ∈ Gsuch that vg = v for all v ∈ V . In particular, V is the trivial irreducible module if

and only if ker ρ = G. On the other hand, a faithful module is one that affords a

faithful representation. The following result is well-known.


Proposition 2.8.4. Let G be a simple group, and let V be a G-module such that

G does not act trivially on V . Then V is faithful. In particular, each irreducible

G-module of dimension at least 2 is faithful.

Proof. Let ρ be the representation afforded by V . Since ker ρ C6 G with G simple,

we have ker ρ ∈ 1, G. If this kernel is G, then G acts trivially on V , which is not

the case. Hence ker ρ = 1, i.e., V is faithful. Now, suppose that V is irreducible

with dim(V ) > 2. If G acts trivially on V , then G stabilises each 1-dimensional

subspace of V . This contradicts the irreducibility of V , which means that G does

not act trivially on V , and thus V is faithful.

We recall that if V is a vector space over F, then the vectors of the dual vector

space V ∗ are the linear maps from V to F.

Proposition 2.8.5 ([13, Ch. 1.8.1]). Let V be an F[G]-module. Then V ∗ is also

an F[G]-module, called the dual module (or contragredient module) of V . Here, for

g ∈ G and φ ∈ V ∗, φg is the linear map in V ∗ that maps each v ∈ V to (vg−1

)φ.

If we extend the map g 7→ g−1 linearly in F[G], then the action of G on V ∗ also

extends linearly in F[G].

Proposition 2.8.6 ([12, p. 235]). Let U and V be isomorphic modules. Then U∗

and V ∗ are also isomorphic modules.

Proposition 2.8.7 ([12, p. 240]). Let V be a (finite-dimensional) module. Then

(V ∗)∗ ∼= V as modules.

Proposition 2.8.8 ([13, Ch. 1.8.2]). Let α ∈ Aut(G), and let V be an F[G]-module

with associated map f : V × F[G]→ V . Then we obtain an F[G]-module V α by

twisting V by α. Here, the map fα : V α×F[G]→ V α associated with V α is defined

by (v, g)fα := (v, gα)f for each v ∈ V and g ∈ G, and this extends linearly in F[G].

Note that V and V α are equal as vector spaces, and hence they have the same

dimension.

Corollary 2.8.9. If U is the set of submodules of V , then Uα | U ∈ U is the

set of submodules of V α, with distinct submodules of V corresponding to distinct

submodules of V α.

Proof. Let f and fα be as in Proposition 2.8.8, and let U ∈ U . Then (u, g)f ∈ Ufor all u ∈ U and all g ∈ G. Since gα ∈ G, we have (u, gα)f = (u, g)fα ∈ U for all

u ∈ U and all g ∈ G, and hence Uα is a submodule of V α. Thus Uα | U ∈ U is

a subset of the submodules of V α, with no two submodules of V corresponding to

the same submodule of V α. Applying this argument with U replaced by Uα and Uα

replaced by (Uα)α−1

= U shows that each submodule of V α lies in Uα | U ∈ U.


Thus V and V α have equivalent submodule structures (in terms of containments

and dimensions).

Definition 2.8.10 ([13, p. 39]). Let V1 and V2 be F[G]-modules. Then V1 and V2

are quasi-equivalent and belong to the same quasi-equivalence class if V1∼= V α

2 for

some α ∈ Aut(G).

Note that quasi-equivalence is an equivalence relation on the set of F[G]-modules,

and that isomorphic modules are also quasi-equivalent.

Lemma 2.8.11 ([13, p. 39–40]). Let d be a positive integer, and let V1 and V2 be

d-dimensional F[G]-modules. If V1 and V2 are quasi-equivalent, then the images

of the afforded representations are conjugate in GL(d,F). Furthermore, if V1 and

V2 are faithful, and if the images of their afforded representations are conjugate in

GL(d,F), then V1 and V2 are quasi-equivalent.

We now define two important classes of modules.

Definition 2.8.12 ([17, p. 17]). A module V is called uniserial if each nonzero

submodule of V contains exactly one maximal submodule.

Equivalently, a uniserial module has exactly one composition series. It is clear

that each irreducible module is uniserial. For a reducible example, let V be the

2-dimensional module for the cyclic group Z2 over the field F2, where the involution

in Z2 interchanges the basis vectors e1 and e2 of V . Then the subspace of V spanned

by e1 + e2 is the unique proper nonzero submodule of V , and hence V is uniserial.

Definition 2.8.13 ([4, p. 38]). A nonzero module is semisimple if it is a direct sum

of irreducible submodules.

In particular, each irreducible module is semisimple. If V is the 2-dimensional

F3[Z2]-module, where the involution in Z2 interchanges the basis vectors e1 and e2

of V , then the subspace spanned by e1 + e2 and the subspace spanned by e1− e2 are

1-dimensional submodules of V that intersect trivially. Thus V is the direct sum of

these irreducible submodules, and so V is a reducible semisimple module.

Lemma 2.8.14 ([13, Lemma 1.8.11]). Let V :=⊕

U∈U U be an F[G]-module, with

U a set of pairwise non-isomorphic irreducible submodules of V . Then U is the set

of irreducible submodules of V .

The following definition applies to the module V from the above lemma.

Definition 2.8.15 ([20, p. 79]). A (semisimple) module is called multiplicity free if

it is a direct sum of pairwise non-isomorphic irreducible submodules.


Corollary 2.8.16. Let V be a multiplicity free module, and let U be the set of

irreducible submodules of V . Then ⊕

X∈X X | X ⊆ U is the set of submodules of

V , with distinct subsets X of U corresponding to distinct submodules of V .

Proof. As V is multiplicity free, Lemma 2.8.14 gives V =⊕

U∈U U . Thus for each

X ⊆ U , we have∑

X∈X X =⊕

X∈X X, which is a submodule of V . Moreover,

distinct subsets of U correspond to distinct subspaces of V , and hence to distinct

submodules. In particular, the empty subset corresponds to the zero submodule.

Finally, each submodule W of V is semisimple [4, p. 39], and in fact multiplicity

free because its set of irreducible submodules is a subset X of U , in which no two

submodules are isomorphic. Therefore, W =⊕

X∈X X by Lemma 2.8.14.

In the following proposition, and throughout this thesis, when we refer to “the

composition factors” of a module V , we mean the multiset of its composition factors.

For example, if we say that V has exactly two composition factors, then each com-

position series for V has the form 0 = V0 ⊂ V1 ⊂ V2 = V . However, it may be the

case that V2/V1∼= V1/V0, so that the two composition factors of V are isomorphic.

Proposition 2.8.17. Let V be a module. Then the following are equivalent.

(i) The irreducible submodules of V are pairwise non-isomorphic, with the sum

of their dimensions equal to dim(V ).

(ii) The composition factors of V are pairwise non-isomorphic, with each isomor-

phic to an irreducible submodule of V .

(iii) V is multiplicity free.

Proof. Observe that each irreducible submodule of V is isomorphic to a composition

factor of V , and that dim(V ) is equal to the sum of the dimensions of the composition

factors of V . First, suppose that the irreducible submodules of V are pairwise non-

isomorphic, with the sum of their dimensions equal to dim(V ). Then isomorphism

of modules gives a 1-1 correspondence between the irreducible submodules of V and

the composition factors of V . Thus (i) implies (ii).

Suppose now that the composition factors of V are pairwise non-isomorphic,

with each isomorphic to an irreducible submodule of V . Let U1, . . . , Un be a set of

irreducible submodules of V , such that each composition factor of V is isomorphic to

a unique submodule in this set. In addition, let Vi :=∑i

j=1 Uj for each i ∈ 1, . . . , n.Observe that, for a given i, Vi is a submodule of V , and if Vi =

⊕ij=1 Uj, then Vi

is multiplicity free. We will show that each Vi is indeed equal to this direct sum,

by induction on i. First, V1 = U1 by definition. For the inductive step, assume

that Vi =⊕i

j=1 Uj for some i ∈ 1, . . . , n− 1. Then Vi is multiplicity free, and so

U1, . . . , Ui is the set of irreducible submodules of Vi by Lemma 2.8.14. This means

that Ui+1 is not a submodule of Vi. Moreover, Vi∩Ui+1 is a submodule of Ui+1, which


is irreducible, and hence this intersection is trivial. Thus Vi+1 =∑i+1

j=1 Uj =⊕i+1

j=1 Uj.

By induction, each Vi is multiplicity free. As dim(Vn) is equal to the sum of the

dimensions of the composition factors of V , which is equal to dim(V ), we have

V = Vn, which is multiplicity free. Hence (ii) implies (iii).

Finally, suppose that V is multiplicity free. We have from Lemma 2.8.14 that

V is equal to the direct sum of its irreducible submodules, which are pairwise non-

isomorphic. Thus the sum of the dimensions of these irreducible submodules is equal

to dim(V ). Therefore, (iii) implies (i).

Now, if U and V are modules, then (u, 0) | u ∈ U is a submodule of U⊕V that

we can identify with U , and (0, v) | v ∈ V is a submodule of U ⊕ V that we can

identify with V . It follows that, for each submodule M of U and each submodule

N of V , M ⊕N is a submodule of U ⊕ V . Additionally, the map (u, v) 7→ (v, u) is

an F[G]-isomorphism from U ⊕ V to V ⊕ U .

Proposition 2.8.18 ([65, p. 126]). Let U and V be F[G]-modules, and let M and N

be submodules of U and V , respectively. Then the F[G]-modules (U ⊕V )/(M ⊕N)

and (U/M)⊕ (V/N) are isomorphic.

Proof. Consider the map φ : (U ⊕ V )/(M ⊕ N) → (U/M) ⊕ (V/N) that maps

(u, v) + M ⊕ N to (u + M, v + N) for each u ∈ U and v ∈ V . Observe that if

u1, u2 ∈ U and v1, v2 ∈ V , then (u1, v1)− (u2, v2) = (u1−u2, v1−v2) ∈M⊕N if and

only if u1−u2 ∈M and v1−v2 ∈ N . Therefore, (u1, v1)+M⊕N = (u2, v2)+M⊕Nif and only if u1 +M = u2 +M and v1 +N = v2 +N . Thus φ is both well-defined and

injective. It is also clear that φ is surjective and linear, and hence it is a bijective

linear transformation. Finally, let x ∈ F[G], u ∈ U and v ∈ V . Then

(((u, v) +M ⊕N)x)φ = ((ux, vx) +M ⊕N)φ = (ux +M, vx +N)

= (u+M, v +N)x = (((u, v) +M ⊕N)φ)x.

Therefore, φ is an F[G]-isomorphism, and so (U⊕V )/(M⊕N) ∼= (U/M)⊕(V/N).

The following result is our own.

Theorem 2.8.19. Let U and V be F[G]-modules with no common composition

factors. Then the set of submodules of U⊕V is the set of direct sums of submodules

of U and submodules of V .

Proof. As each direct sum of a submodule of U and a submodule of V is indeed

a submodule of U ⊕ V , it suffices to show that these are the only submodules. We

prove this by induction on dim(U⊕V ). If dim(U) = 0 (respectively, if dim(V ) = 0),

then U⊕V is equal to V (respectively, U), and hence the result holds. In particular,

the result holds whenever dim(U ⊕ V ) 6 1. For the inductive step, suppose that

there exists a positive integer n such that the result holds whenever dim(U⊕V ) 6 n.


Consider the case where dim(U ⊕ V ) = n + 1. We may assume that dim(U)

and dim(V ) are positive. Suppose that K is a submodule of U ⊕ V that is not

the direct sum of a submodule of U and a submodule of V . In addition, let N be

a maximal submodule of V . Then U ⊕ N is a proper submodule of U ⊕ V , and

there are no common composition factors between U and N . It follows from the

inductive hypothesis that K ∩ (U ⊕N) = X ⊕ Y , for some submodule X of U and

some submodule Y of V . In particular, K contains X ⊕ Y , and K/(X ⊕ Y ) is a

submodule of (U ⊕ V )/(X ⊕ Y ) by the Correspondence Theorem.

Now, (U ⊕ V )/(X ⊕ Y ) ∼= (U/X)⊕ (V/Y ) by Proposition 2.8.18. Observe that

U/X and V/Y have no common composition factors, and that if either dim(X) or

dim(Y ) is positive, then dim(U/X⊕V/Y ) < dim(U⊕V ). In this case, the inductive

hypothesis implies that each submodule of (U/X) ⊕ (V/Y ) is the direct sum of a

submodule of U/X and a submodule of V/Y . It follows that each submodule of

(U ⊕V )/(X⊕Y ) can be written as (A⊕B)/(X⊕Y ), where A is a submodule of U

that contains X and B is a submodule of V that contains Y . The Correspondence

Theorem therefore implies that there exist such A and B with A ⊕ B = K. This

contradicts the assumption that K is indecomposable. Hence dim(X) = dim(Y ) =

0, i.e., K∩(U⊕N) is trivial. By symmetry of U and V , if M is a maximal submodule

of U , then M ⊕ V is a proper submodule of U ⊕ V , and K ∩ (M ⊕ V ) is trivial. In

particular, (U ⊕N) +K = (U ⊕N)⊕K and (M ⊕ V ) +K = (M ⊕ V )⊕K.

Finally, we have from Proposition 2.8.18 that (U ⊕ V )/(U ⊕ N) ∼= V/N and

(U ⊕ V )/(M ⊕ V ) ∼= U/M . As N and M are maximal submodules of V and U ,

respectively, it follows that U ⊕N and M ⊕ V are maximal submodules of U ⊕ V .

Hence (U⊕N)⊕K = U⊕V = (M⊕V )⊕K. Thus the Second Isomorphism Theorem

gives (U ⊕ V )/(U ⊕ N) ∼= K ∼= (U ⊕ V )/(M ⊕ V ). We therefore have from the

previous isomorphisms that V/N ∼= U/M . However, these are composition factors

of V and U , respectively, which do not have any common composition factors. This

is a contradiction, and thus the submodule K does not exist. The result therefore

follows by induction.

It is immediate that if W is a finite direct sum of modules, no two of which have

common composition factors, then the set of submodules of W is the set of direct

sums of submodules of the direct summands of W . In particular, this holds when

W is multiplicity free, and so Corollary 2.8.16 follows from the above theorem and

Lemma 2.8.14.

The next result shows that the conclusion of the above theorem does not hold

for all direct sums of modules.

Proposition 2.8.20. Let U and V be nonzero F[G]-modules, suppose that there ex-

ists a surjective F[G]-homomorphism φ : U → V , and let M := (u, (u)φ) | u ∈ U.Then M is a submodule of U ⊕ V that is isomorphic to U . Furthermore, M cannot


be written as the direct sum of a submodule of U and a submodule of V , and M

does not contain V .

Proof. Since φ is linear, and since U and V are subspaces of U ⊕ V , it is easy

to see that M is also a subspace of U ⊕ V . Moreover, if g ∈ G and u ∈ U , then

(u, (u)φ)g = (ug, ((u)φ)g) = (ug, (ug)φ), since φ is an F[G]-homomorphism. As

U is an F[G]-module, we have ug ∈ U , and hence (ug, (ug)φ) ∈ M . Thus M is a

submodule of U⊕V . Furthermore, the map ψ : U →M defined by (u)ψ := (u, (u)φ)

for each u ∈ U is an F[G]-isomorphism, and so M ∼= U .

Now, each vector in the submodule V of U ⊕ V has a first coordinate of 0, and

hence M ∩ V = (0, 0). Thus M does not contain V . It also follows that if M can

be written as the direct sum of a submodule of U and a submodule of V , then this

submodule of V is the zero submodule, i.e., M is a submodule of U . As M ∼= U ,

this implies that M = U . However, each vector in the submodule U of U ⊕ V has a

second coordinate of 0, whereas there exists a vector (u, (u)φ) ∈ M with (u)φ 6= 0.

This is a contradiction, and therefore M cannot be written as the direct sum of a

submodule of U and a submodule of V .

Proposition 2.8.21 ([19, p. 69–70]). Let U and V be F[G]-modules. The tensor

product U ⊗V is an F[G]-module. Here, we define (u⊗ v)g := ug⊗ vg for all u ∈ U ,

v ∈ V and g ∈ G, and we extend this linearly in U ⊗ V and in F[G].

Note that the above proposition readily extends to tensor products of any finite

number of modules.

Proposition 2.8.22 ([70, p. 71]). Let V be an F[G]-module, and let k > 2 be

an integer. The exterior power AkV is an F[G]-module. Here, (v1 ∧ · · · ∧ vk)g :=

v1g ∧ · · ·∧ vkg for all v1, . . . , vk ∈ V and g ∈ G, and this extends linearly in AkV and

in F[G].

It is easy to see that if U and V are isomorphic F[G]-modules, then AkU and

AkV are also isomorphic F[G]-modules for each integer k > 2. The remaining results

in this section are our own.

Proposition 2.8.23. Let V1 and V2 be F[G]-modules, let α ∈ Aut(G), and let k > 2

be an integer. Then V α1 ⊗ V α

2 and (V1 ⊗ V2)α are equal as modules, as are Ak(V α1 )

and (AkV1)α. In addition, for each submodule U of V1, the modules V α1 /U

α and

(V1/U)α are equal.

Proof. First, since modules twisted by automorphisms are equal as vector spaces,

we have V α1 ⊗ V α

2 = V1 ⊗ V2 = (V1 ⊗ V2)α, Ak(V α1 ) = AkV1 = (AkV1)α, and

V α1 /U

α = V1/U = (V1/U)α as vector spaces. For i ∈ 1, 2, let fi : Vi × F[G] → Vi

and fiα : V αi × F[G]→ V α

i be the maps associated with the modules Vi and V αi ,


respectively. The definitions of the action of G on each of V α1 ⊗ V α

2 and (V1 ⊗ V2)α

imply that, for u ∈ V1 = V α1 , v ∈ V2 = V α

2 , and g ∈ G,

(u⊗ v)g = ug ⊗ vg = (u, gα)f1 ⊗ (v, gα)f2.

Since these vector spaces are spanned by u ⊗ v | u ∈ V1, v ∈ V2, the linearity of

f1 and f2 implies that V α1 ⊗ V α

2 = (V1⊗ V2)α as modules. A similar argument using

Lemma 2.7.3 and the definition of the action of G on an exterior power of a module

shows that Ak(V α) = (AkV )α as modules. Finally, the vectors of each of V α1 /U

α

and (V1/U)α are v + Uα | v ∈ V α1 = v + U | v ∈ V1. If v ∈ V1 and g ∈ G, then

with respect to the action of G on each of these modules, we have

(v + U)g = vg + U = (v, gα)f1 + U.

The linearity of f1 implies that V α1 /U

α = (V1/U)α as modules.

Proposition 2.8.24. Let V be a nonzero F[G]-module such that the image of the

afforded representation does not intersect trivially with Z(GL(V )). If X and Y are

submodules of A2V with Y ⊆ X, then V is not isomorphic as an F[G]-module to

the quotient module X/Y . In particular, if F 6= F2 and if V is the faithful module

associated with the natural linear action of G = GL(V ), then V and A2V are not

isomorphic as F[G]-modules.

Proof. As the image of the representation afforded by the F[G]-module V does not

intersect trivially with Z(GL(V )), there exists an element g ∈ G and a nonidentity

element z ∈ Z(GL(V )) such that vg = vz for all v ∈ V . Since Z(GL(V )) can be

identified with Z(GL(d,F)), which consists of the nonzero scalar matrices, we have

vg = λv for all v ∈ V , where λ ∈ F \ 0, 1. Now, for all v1, v2 ∈ V , we have

(v1 ∧ v2)g = vg1 ∧ vg2 = (λv1) ∧ (λv2).

This is equal to λ2(v1 ∧ v2) by the bilinearity of the wedge product. It follows from

Lemma 2.7.3 and the linearity of the action of G on A2V that ug = λ2u for all

u ∈ A2V . Furthermore, if x ∈ X, then

(x+ Y )g = xg + Y = λ2x+ Y = λ2(x+ Y ).

Suppose now that θ : V → X/Y is a linear transformation, and let v ∈ V \ 0.Then (vg)θ = (λv)θ = λ(v)θ, and ((v)θ)g = λ2(v)θ, since (v)θ = x + Y for some

x ∈ X. As λ /∈ 0, 1, we have λ 6= λ2, and hence (vg)θ 6= ((v)θ)g. Thus there

is no F[G]-isomorphism from V to X/Y , i.e., V 6∼= X/Y . Finally, if F 6= F2, then

Z(GL(V )) is nontrivial, and hence the natural F[GL(V )]-module V is not isomorphic

to (A2V )/0 = A2V .

Of course, V and A2V are not even isomorphic as vector spaces unless dim(V ) =

dim(A2V ), which occurs when d = 3 by Lemma 2.7.3. In fact, a computation in

Magma, using the code given in Appendix A.2, shows that V and A2V are not

isomorphic as F[GL(V )]-modules even when F = F2 and V = F32.


2.9 Representation theory: subgroups, subfields and exten-

sion fields

We recall that if H is a subgroup of G, then we can restrict an F[G]-module V to

an F[H]-module V |H , associated with the linear action of H on V . In this section,

we discuss these restricted modules, as well as F[G]-modules viewed as modules over

subfields or extension fields of F.

First, observe that if W is a submodule of a G-module V , then W |H is a sub-

module of V |H . However, if V contains a subspace U such that uy ∈ U for all u ∈ Uand all y ∈ F[H], and such that there exist vectors u ∈ U and x ∈ F[G] such that

ux /∈ U , then the restriction of U to H is a submodule of the restriction of V to H,

but U is not a submodule of V . Thus restricting a module may result in the creation

of new submodules, but never in the suppression of old ones. We sometimes refer

to the submodules of the H-module V |H as H-submodules of V .

Proposition 2.9.1. Let V be a semisimple G-module, and let H be a subgroup

of G. Suppose also that the restriction to H of each composition factor of V is a

composition factor of V |H . Then V |H is semisimple. If V is multiplicity free, and if

non-isomorphic composition factors of V remain non-isomorphic when restricted to

H, then V |H is also multiplicity free.

Proof. As V is semisimple, Definition 2.8.13 implies that the vector space V = V |His equal to the direct sum of some of the irreducible submodules of V . Each of these

irreducible submodules is isomorphic to a composition factor of V , and hence its

restriction to H is an irreducible submodule of V |H . Thus V |H is semisimple. If

V is multiplicity free, and if non-isomorphic composition factors of V remain non-

isomorphic when restricted to H, then the irreducible submodules of V |H are also

pairwise non-isomorphic. Therefore, in this case, V |H is multiplicity free.

We recall that, for a prime p, a Sylow p-subgroup of a finite group G is a subgroup

whose order is the largest power of p that divides |G|. Moreover, Sylow’s Theorems

imply that G contains at least one Sylow p-subgroup.

Lemma 2.9.2 ([4, p. 39]). Let V be a module for a finite group G over a field with

prime characteristic p. Suppose that U is a submodule of V , and that P is a Sylow

p-subgroup of G such that V |P = U |P ⊕W for some submodule W of V |P . Then

there exists a submodule X of V such that V = U ⊕X.

Corollary 2.9.3. Let H be a subgroup of a finite group G, and let p be a prime

such that |G : H| is coprime to p. In addition, let V be a G-module over a field of

characteristic p such that no two submodules of V |H are equidimensional. If U is

a submodule of V such that V |H = U |H ⊕W for some submodule W of V |H , then

there exists a submodule X of V such that W = X|H .

2.9. Representation theory: subgroups, subfields and extension fields 31

Proof. As |G : H| is coprime to p, H contains a Sylow p-subgroup P of G. Thus

U |P and W |P are submodules of V |P , and V |P = U |P ⊕W |P . Lemma 2.9.2 there-

fore implies that there exists a submodule X of V such that V = U ⊕ X. Hence

U |H ⊕W = V |H = U |H ⊕X|H . As no two submodules of V |H are equidimensional,

it follows that X|H = W .

Proposition 2.9.4 ([44, p. 79]). Let V be a G-module, with N a normal subgroup

of G, and let U be a submodule of V |N . For each g ∈ G, U g is a submodule of V |N .

If U is an irreducible submodule of V |N , then so is U g.

Since G acts linearly on V , it is clear that dim(U g) = dim(U).

Corollary 2.9.5. Let N be a normal subgroup of G, and let V be a G-module such

that V |N is multiplicity free, with no two irreducible submodules of V |N having

equal dimension. Then V is multiplicity free, and each submodule of V |N is the

restriction to N of a submodule of V .

Proof. Let U1, . . . , Un be the set of irreducible submodules of V |N . Proposition

2.9.4 implies that, for each g ∈ G and each i ∈ 1, . . . , n, U gi is an irreducible

submodule of V |N with the same dimension as Ui. As Ui is the only such submodule,

Ui = U gi . Therefore, Ui is an irreducible G-submodule of V . Since V |N is multiplicity

free, V is the direct sum of the irreducible G-submodules U1, . . . , Un. If any two of

these G-submodules are isomorphic, then so are their restrictions to H. This is

not the case, and so V is multiplicity free. The description of the submodules of

a multiplicity free module given in Corollary 2.8.16 implies that the submodules of

V |N are as required.

Theorem 2.9.6 (Clifford’s Theorem [44, Theorem 6.5]). Let V be an irreducible

G-module, let N be a normal subgroup of G, and let U be an irreducible submodule

of V |N . Then there exists a nonempty subset S of G such that V |N is the direct

sum of the equidimensional irreducible N -submodules U s | s ∈ S.

For the proof of the following corollary, note that if U and W are submodules of a

G-module V with U ⊆ W , and if H is a subgroup of G, then (W/U)|H = W |H/U |Has modules.

Corollary 2.9.7. Let N be a normal subgroup of G, and let V be a G-module such

that no two submodules of V |N have equal dimension. Then each composition series

for V restricts to a composition series for V |N .

Proof. Let 0 = V0 ⊂ V1 ⊂ · · · ⊂ Vn = V be a composition series for V , and

let i ∈ 1, . . . , n. Then Vi/Vi−1 is an irreducible G-module. As no two sub-

modules of V |N are equidimensional, the Correspondence Theorem implies that

no two submodules of Vi|N/Vi−1|N are equidimensional. It follows from Clifford’s


Theorem that Vi|N/Vi−1|N = (Vi/Vi−1)|N is an irreducible N -module. Therefore,

0 = V0|N ⊂ V1|N ⊂ · · · ⊂ Vn|N = V |N is a composition series for V |N .

Proposition 2.9.8 ([4, p. 118–119]). Let F ⊆ K be a field extension, let V be an

F[G]-module with basis B, and let ρ be the F-representation of G afforded by V .

Then ρ induces a K-representation ρK of G, with the same dimension as ρ. Here, for

each g ∈ G, we can define (g)ρK as the matrix (g)ρ with each entry interpreted as

an element of K. Furthermore, the K[G]-module that affords ρK is V K := K ⊗ V ,

with K considered as a vector space over F. Here, 1⊗ e | e ∈ B is a basis for V K ,

and (α⊗ v)g = α⊗ vg for all α ∈ K, v ∈ V and g ∈ G, with this extending linearly

in F[G].

The F[G]-module V can therefore be considered as a K[G]-module by appropri-

ately “extending” the scalars. Observe that ker ρ = ker ρK , and that dim(V K) =

dim(V ). With a slight abuse of notation, we consider α ⊗ v as αv, for each α ∈ Kand v ∈ V , and thus we consider B as a basis for V K .

Proposition 2.9.9 ([5, p. 113]). Let F ⊆ K be a field extension, and let V be an

F[G]-module with submodule W . Then WK is a submodule of V K . Consequently,

if V K is irreducible, then so is V .

Proof. Let C be a basis for W . By Proposition 2.9.8, C is a K-basis for WK .

Hence WK is a subspace of V K . Moreover, since W is a submodule of V , we have

eg ∈ W for each e ∈ C and g ∈ G, and we can consider eg as the vector 1Keg ∈ WK .

It follows from the linearity of the map associated with the K[G]-module V K that

WK is a submodule of V K .

Thus, in terms of containments and dimensions, the submodule structure of V K

is obtained by starting with that of V , and possibly inserting new submodules in

between some of the original submodules.

The following lemma is a collection of related well-known results.

Lemma 2.9.10. Let F ⊆ K be a field extension, and let V and U be F[G]-modules

with W a submodule of V , and let n > 2 be an integer. Then, as K[G]-modules,

(V ⊗ U)K = V K ⊗ UK ; (AnV )K = An(V K); and (V/W )K ∼= V K/WK .

Proof. Let B and C be bases for V and U , respectively. Then a basis for V ⊗U is

D := e⊗ f | e ∈ B, f ∈ C. We have from Proposition 2.9.8 that, over K, D is a

basis for both (V ⊗U)K and V K ⊗UK , and hence these are equal as vector spaces.

Now, the linear action of G on each of (V ⊗ U)K and V K ⊗ UK is determined

uniquely by the image of each basis vector under each group element. We have

(e ⊗ f)g = eg ⊗ f g for all e ∈ B, f ∈ C and g ∈ G, and thus the two modules

are equal. The equality (AnV )K = An(V K) follows from a similar proof that uses

Lemma 2.7.3 and the definition of the action of G on each module.


Next, we can choose a basis B for V so that a subset B1 of B is a basis for W .

Then e+W | e ∈ B \B1 is a basis for V/W and for (V/W )K . Additionally, WK is

a submodule of V K by Proposition 2.9.9, and so e+WK | e ∈ B \B1 is a basis for

the quotient module V K/WK . The definition of the action ofG on a quotient module

implies that the linear map from (V/W )K to V K/WK that sends e+W to e+WK

for each e ∈ B \B1 is a K[G]-isomorphism, and hence (V/W )K ∼= V K/WK .

Definition 2.9.11 ([13, p. 38, p. 40]). Let ρ be an F-representation of G. Then ρ

is absolutely irreducible if, for every extension field K of F, the K-representation ρK

is irreducible. Similarly, a G-module V is absolutely irreducible if V K is irreducible

for every extension field K of F, and we say that G acts absolutely irreducibly on V .

We also say that a subgroup of a general linear group is absolutely irreducible if it

is the image of an absolutely irreducible representation.

Note that, since F is an extension field of itself, every absolutely irreducible F-

representation is irreducible. Additionally, since each 1-dimensional F-representation

ρ is irreducible by definition, and since ρK is 1-dimensional for each extension field

K of F, it follows that ρ is absolutely irreducible. In general, however, an irre-

ducible F-representation is not necessarily absolutely irreducible, as we illustrate in

the following example.

Example 2.9.12 ([33, p. 278]). Let ρ be the 2-dimensional R-representation of

the cyclic group Z4 that maps a generator x of Z4 to

(0 1

−1 0

)∈ GL(2,R). No

1-dimensional subspace of R2 is fixed by this matrix, which corresponds to an an-

ticlockwise rotation of 90° about the origin, and hence this representation is irre-

ducible. As C is an extension field of R, ρ induces a C-representation ρC of Z4 that

maps x to

(0 1

−1 0

)∈ GL(2,C). The 1-dimensional subspaces 〈(1, i)〉 and 〈(1,−i)〉

of C2 are fixed by this matrix, since, for example, (1, i)

(0 1

−1 0

)= (−i, 1) =

−i(1, i). These subspaces are therefore fixed by the image of Z4 in GL(2,C), and

hence ρC is reducible. This means that ρ is not absolutely irreducible.

The following definition is given by Kleidman and Liebeck [53, p. 52, p. 192],

but we use the terminology of Bray, Holt and Roney-Dougal [13, p. 41].

Definition 2.9.13. An F[G]-module can be written over a subfield E of F if the

module affords a representation that maps each element of G to a matrix with entries

in E. In this case, we also say that the image of the afforded representation can be

written over E.


Lemma 2.9.14 ([13, p. 40–41]). Let V be a vector space over F, and let G be an

absolutely irreducible subgroup of GL(V ).

(i) If G is perfect and preserves a non-degenerate bilinear or σ-Hermitian form

up to scalars, then G preserves the form absolutely.

(ii) If G preserves a nonzero bilinear (respectively, σ-Hermitian) form β absolutely,

then the only bilinear forms (respectively, σ-Hermitian forms for the same

σ ∈ Aut(F)) that G preserves are the scalar multiples of β, and NGL(V )(G)

preserves β up to scalars.

(iii) If F is a finite field, and if no conjugate of G in GL(V ) can be written over a

proper subfield of F, then G does not preserve both a nonzero bilinear form

and a nonzero σ-Hermitian form.

Definition 2.9.15 ([4, p. 123]). A field F is a splitting field for a finite group G if

every irreducible F-representation of G is absolutely irreducible.

For example, let G be a p-group, let F be a field of characteristic p, and let ρ be

an irreducible Fp-representation of G. Then ρ has dimension 1 [59, Theorem 1.3.11],

and is therefore absolutely irreducible. Thus Fp is a splitting field for G.

Proposition 2.9.16 ([44, Corollary 9.8]). Let F be a splitting field for a finite

group G, let F ⊆ K be field extension, and let Vi be the set of distinct (up to

isomorphism) irreducible F[G]-modules. Then V Ki is the set of distinct irreducible

K[G]-modules.

The remaining results in this section are our own.

Proposition 2.9.17. Let F be a splitting field for a finite group G, let F ⊆ K be

an algebraic extension, and let V be an F[G]-module such that V K is multiplicity

free. Then V is also multiplicity free.

Proof. Since K is algebraic over F, with V K semisimple, V is also semisimple by [5,

Proposition 2.1.5]. Hence V is the direct sum of a set U of irreducible submodules of

V . For each U ∈ U , UK is an irreducible submodule of V K because F is a splitting

field for G. As V K is multiplicity free, the submodules in UK | U ∈ U are pairwise

non-isomorphic. Thus the submodules in U are also pairwise non-isomorphic by

Proposition 2.9.16, and therefore V is multiplicity free.

Lemma 2.9.18. Let F be a splitting field for a finite group G, and let F ⊆ K be a

field extension. In addition, let V be an F[G]-module, and let [Ui] be the multiset of

composition factors of V . Then [UKi ] is the multiset of composition factors of V K .

Proof. Let 0 = V0 ⊂ V1 ⊂ · · · ⊂ Vn = V be a composition series for V . Then

0 = V K0 ⊂ V K

1 ⊂ · · · ⊂ V Kn = V K is a chain of submodules of V K by Proposition


2.9.9. For each i ∈ 1, . . . , n, the composition factor Vi/Vi−1 is irreducible, as

is (Vi/Vi−1)K since F is a splitting field for G. We have V Ki /V

Ki−1∼= (Vi/Vi−1)K

from Lemma 2.9.10, which means that V Ki /V

Ki−1 is irreducible, and hence the given

chain of submodules of V K is a composition series for V K . It follows from the

isomorphism V Ki /V

Ki−1∼= (Vi/Vi−1)K that the multiset of composition factors of V K

is as required.

Lemma 2.9.19. Let F be a splitting field for a finite group G, let F ⊆ K be a field

extension, and let V be an F[G]-module. If V K is uniserial, then V is also uniserial,

with an equivalent submodule structure (in terms of containments and dimensions)

to V K . Furthermore, if K is algebraic over F and if V is uniserial, then V K is also

uniserial, with an equivalent submodule structure to V .

Proof. Throughout this proof, we use the fact that a submodule U of V is irre-

ducible if and only if UK is irreducible, since F is a splitting field for G. Furthermore,

V and V K have the same number of composition factors by Lemma 2.9.18. Thus

if both V and V K are uniserial, i.e., if each has a unique composition series, then

Proposition 2.9.9 implies that the two modules have equivalent submodule struc-

tures. In fact, this proposition implies that if V K has exactly one composition

series, then so does V , i.e., if V K is uniserial, then so is V .

Suppose now that K is algebraic over F and that V is uniserial. By Proposition

2.9.9, V K is uniserial if and only if each submodule of V K is equal to UK for some

corresponding submodule U of V . We will prove that V K is uniserial by induction

on dim(V ). Clearly, V K is uniserial if dim(V ) = 0. For the inductive step, suppose

that V K is uniserial whenever dim(V ) is at most n, for some nonnegative integer n.

Consider the case where dim(V ) = n + 1. If V is irreducible, then V K is also

irreducible and hence uniserial. Otherwise, let M be the unique irreducible sub-

module of V . The quotient module V/M is uniserial of dimension less than n + 1,

and hence the inductive hypothesis implies that (V/M)K is uniserial. Moreover,

(V/M)K ∼= V K/MK by Lemma 2.9.10. It follows from the Correspondence Theorem

that each submodule of V K that contains MK is equal to UK for some corresponding

submodule U of V . Also note that MK is the unique irreducible submodule of V K

that is equal to UK for a corresponding submodule U of V .

Now, for a proof by contradiction, let Y be the smallest submodule of V K such

that, for each submodule U of V , Y 6= UK . Then Y does not contain MK . This

means that Y is irreducible, and that Y ∩MK = 0. Therefore, the submodule

Y + MK of V K is equal to Y ⊕MK . This submodule properly contains MK , and

hence Y ⊕ MK = UK , for some (not necessarily proper) reducible submodule U

of V . As Y and MK are irreducible, UK is semisimple. Moreover, K is algebraic

over F, and hence U is semisimple by [5, Proposition 2.1.5]. However, M is the

unique irreducible submodule of the reducible module U , which means that U is


not semisimple. This is a contradiction, and thus the submodule Y does not exist.

Therefore, V K is uniserial, and the result follows by induction.

2.10 Aschbacher’s Theorem

In this section, we define certain classes of subgroups of classical groups, and then

state the related Aschbacher’s Theorem [3]. Our formulations of these definitions

and this theorem are from Bray, Holt and Roney-Dougal [13, Ch. 2]. Here, V := Fdq ,where q is a power of a prime p, and where d > 1 is an integer.

Definition 2.10.1. Let G be a group that acts linearly on V , and suppose that V

can be decomposed as a direct sum of proper subspaces, or as a tensor product of

lower-dimensional vector spaces. Then we say that G stabilises this decomposition

of V if G permutes the components of the decomposition.

In other words, G stabilises the decomposition if the image of each of its com-

ponents is also a component.

Let (T, S, c) be an element of the set

(GL(d, q), SL(d, q), 2), (Sp(d, q), Sp(d, q), 4),

(GOε(d, q),Ωε(d, q), 7), (GU(d, q1/2), SU(d, q1/2), 3),

where ε ∈ ,+,− and d > c. In addition, letR be a subgroup of T that contains S.

Definition 2.10.2. The geometric subgroups of R are the subgroups that belong to

(at least) one of the following classes.

C1 : Stabilisers of certain nontrivial proper subspaces of V .

C2 : Stabilisers of direct sum decompositions of V into proper equidimensional

subspaces.

C3 : Stabilisers of extension fields Fqr of Fq, for primes r dividing d.

C4 : Stabilisers of tensor product decompositions of V into two lower-dimensional

vector spaces.

C5 : Stabilisers of subfields Fq1/r of Fq, for primes r.

C6 : Normalisers of symplectic-type or extraspecial r-subgroups of R, for primes

r 6= p such that d is a power of r.

C7 : Stabilisers of tensor product decompositions of V into equidimensional vector

spaces smaller than V .

C8 : The largest groups of similarities of non-degenerate σ-Hermitian or reflexive

bilinear forms (when T = GL(d, q)), or the largest groups of isometries of

non-degenerate quadratic forms (when T = Sp(d, q) with q even).

2.10. Aschbacher’s Theorem 37

Note that the component vector spaces in the decomposition of V associated

with C4-subgroups may or may not be equidimensional. Moreover, it is clear that

C7-subgroups are only defined when d = mn for positive integers m < d and n > 1.

Definition 2.10.3. A subgroup H of R lies in class C9 if all of the following hold:

(i) H/(H ∩ Z(GL(d, q))) is almost simple;

(ii) H does not contain S;

(iii) H∞ is absolutely irreducible;

(iv) no conjugate of H∞ in GL(d, q) can be written over a proper subfield of Fq;

(v) H∞ preserves no nonzero σ-Hermitian or reflexive bilinear form if T = GL(d, q);

(vi) H∞ preserves no nonzero σ-Hermitian or quadratic form if T = Sp(d, q) and

q is even;

(vii) H∞ preserves no nonzero σ-Hermitian or symmetric form if T = Sp(d, q) and

q is odd;

(viii) H∞ preserves no nonzero σ-Hermitian or alternating form if T = GOε(d, q);

and

(ix) H∞ preserves no nonzero reflexive bilinear form if T = GU(d, q1/2).

The quotient group H/(H∩Z(GL(d, q))) is the projective version of H by Defini-

tion 2.6.11. Also observe that if H is non-abelian and simple, and therefore perfect,

then we have H/(H ∩ Z(GL(d, q))) = H/1 = H and H∞ = H. Note that Bray, Holt

and Roney-Dougal write S to denote the Aschbacher class C9, but we use notation

consistent with Bamberg et al. [7]. The subgroup classes C1, . . . , C9 of R are the

Aschbacher classes of R.

Proposition 2.10.4 ([13, p. 56]). Let H be a subgroup of R such that H∞ is

absolutely irreducible and such that no conjugate of H∞ can be written over a

proper subfield of Fq. Then H does not lie in any C1-, C3- or C5-subgroup of R. In

particular, this holds if H is a C9-subgroup of R.

Proof. Suppose that X ∈ C1 ∪ C3 ∪ C5, with H 6 X. Then H∞ 6 X∞ 6 X. Since

H∞ is absolutely irreducible, and since no conjugate of H∞ can be written over a

proper subfield of Fq, the same must hold for X∞ and X. It follows from Definition

2.10.2 that X /∈ C1. If X ∈ C3, then X∞ is not absolutely irreducible [13, p. 56],

and hence X /∈ C3. Finally, if X ∈ C5, then there exists an element g ∈ GL(d, q)

such that (Xg)∞ can be written over a proper subfield of Fq [13, p. 69]. The group

(Xg)∞ is equal to the conjugate (X∞)g of X∞ by Lemma 2.1.8, and hence X /∈ C5.

Thus there is no group X ∈ C1 ∪C3 ∪C5 with H 6 X. Definition 2.10.3 implies that

the necessary conditions for H hold for each C9-subgroup of R.


Theorem 2.10.5 (Aschbacher’s Theorem). Suppose that a maximal subgroup H

of R does not contain S and is not a geometric subgroup of R. Then H ∈ C9.

In this thesis, when we say “a maximal Ci-subgroup of R”, we mean a maximal

subgroup of R that is also a Ci-subgroup of R. All maximal subgroups of S have

been classified by Bray, Holt and Roney-Dougal [13] for d 6 12. Note that Kleidman

[52] previously presented a classification of the maximal geometric subgroups of S

for d 6 12, but without proof. Additionally, Kleidman and Liebeck [53] classified

the maximal geometric subgroups for all d > 12, while Schroder [68] classified the

maximal C9-subgroups for d ∈ 13, 14, 15. However, there is no known method of

classifying the maximal C9-subgroups uniformly for all d [13, p. 2].

Recall that for a subgroup K 6 GL(d, q), K denotes Z(GL(d, q))K/Z(GL(d, q)),

which is a subgroup of PGL(d, q) by the Correspondence Theorem. The following

lemma is our own.

Lemma 2.10.6. Let H be a C9-subgroup of R, and let J := H/(H ∩ Z(GL(d, q))).

Then H∞ is quasisimple, with H∞ ∼= H∞/Z(H∞) isomorphic to a subgroup of

PGL(d, q).

Proof. The group J is almost simple by Definition 2.10.3, and hence X C6 J 6

Aut(X) for some non-abelian simple group X. Let Z0 := H∞ ∩ Z(GL(d, q)). We

have X = J∞ by Proposition 2.2.2; J∞ ∼= H∞ and H∞ ∼= H∞/Z0 by the Second

Isomorphism Theorem; and H∞ = H∞ by Lemma 2.1.8. Thus X is isomorphic to

the subgroup H∞ of PGL(d, q), and also to H∞/Z0. As X is simple, the Correspon-

dence Theorem implies that the only normal subgroups of H∞ that contain Z0 are

H∞ and Z0. Furthermore, as X is non-abelian, so is H∞, and thus Z(H∞) < H∞.

Since the normal subgroup Z(H∞) of H∞ contains Z0, it follows that Z(H∞) = Z0.

Thus the quotient of the perfect group H∞ by its centre is the non-abelian simple

group X, and hence H∞ is quasisimple by Definition 2.2.3.

2.11 Lie powers of vector spaces

We now introduce Lie powers of vector spaces. In the next chapter, we will see

that these Lie powers are involved in the construction of certain important p-groups.

First, we require the following definition. Here, and in subsequent definitions in

this section, we use notation consistent with Bamberg et al. [7].

Definition 2.11.1 ([50, p. 1]). A Lie algebra is a vector space L equipped with a

bilinear map [·, ·] from L× L to L such that, for all x, y, z ∈ L:

(i) The alternating property: [x, x] = 0.

(ii) The Jacobi identity: [[x, y], z] + [[y, z], x] + [[z, x], y] = 0.

2.11. Lie powers of vector spaces 39

We call the operation [·, ·] a Lie bracket [23, p. 1]. Note that we can write

(L, [·, ·]) to denote the Lie algebra L equipped with the Lie bracket [·, ·]. As its

name suggests, L is indeed an algebra, with xy := [x, y] for all x, y ∈ L. In fact, it

is easy to see that if V is an algebra whose multiplication is associative, then V is a

Lie algebra when equipped with the Lie bracket defined by [u, v] := uv − vu for all

u, v ∈ V [67, p. 2]. The resulting Lie algebra may be non-associative, for example

when V is the associative algebra of matrices mentioned below Definition 2.7.6.

Observe that the bilinearity and alternating property of the Lie bracket imply

that 0 = [x + y, x + y] = [x, x] + [x, y] + [y, x] + [y, y] = [x, y] + [y, x]. Hence we

obtain the following.

Proposition 2.11.2 ([50, p. 1]). Let (L, [·, ·]) be a Lie algebra, and let x, y ∈ L.

Then [y, x] = −[x, y].

Now, let V be a nonzero vector space over the field F. The following definition

involves the tensor algebra T (V ) and its associated bilinear multiplication, as defined

in Definition 2.7.7. Recall that we can identify the tensor powers of V with certain

subspaces of T (V ).

Definition 2.11.3 ([7, §2]). We define L(V ) to be the smallest subspace of T (V )

that contains V and that is closed under the bracket operation [u, v] := uv − vu for

u, v ∈ L(V ), where uv the product in T (V ).

Since T (V ) is an associative algebra, we obtain the following.

Proposition 2.11.4 ([7, §2]). The subspace L(V ) of T (V ) is a Lie algebra when

equipped with the bracket operation [·, ·].

In fact, (L(V ), [·, ·]) is called the free Lie algebra on V [49, §1]. When applying

the associated Lie bracket [·, ·] recursively, we often use left-normed notation. For

example, [v1, v2, v3, v4] := [[[v1, v2], v3], v4] for v1, v2, v3, v4 ∈ L(V ).

Definition 2.11.5 ([49, §1]). Let k be a positive integer. Then the k-th Lie power

of V is the subspace of T (V ) defined by LkV := T kV ∩ L(V ).

In particular, since L(V ) contains V = T 1V , we have L1V = V .

The next result follows from the definitions of T (V ), L(V ) and Lie powers of V ;

the definition of tensor products of multiple vectors; and our identification between

tensor powers of V and certain subspaces of T (V ).

Proposition 2.11.6. If u ∈ LjV and v ∈ LkV for positive integers j and k, then

[u, v] ∈ Lj+kV . Furthermore, for each integer k > 2, [v1, . . . , vk] | v1, . . . , vk ∈ V is a spanning set for LkV .


Note that if u ∈ LjV and v ∈ LkV for positive integers j and k, then Definitions

2.11.3 and 2.11.5 allow us to write [u, v] = u ⊗ v − v ⊗ u, with u and v considered

as tensors in T jV and T kV , respectively. Now, Proposition 2.8.21 defines a linear

action of GL(V ) on T kV . If α ∈ GL(V ) and v1, . . . , vk ∈ V , then the image of

[v1, . . . , vk] ∈ LkV ⊆ T kV under α is [vα1 , . . . , vαk ], which is also a vector in LkV .

Hence the linear action of GL(V ) on T kV induces a linear action of GL(V ) on LkV .

We summarise this in the following proposition.

Proposition 2.11.7 ([49, p. 2]). Let k > 2 be an integer. Then GL(V ) acts linearly

on LkV , with [v1, . . . , vk]α := [vα1 , . . . , v

αk ] for all v1, . . . , vk ∈ V and all α ∈ GL(V ).

It follows immediately that LkV is a GL(V )-module associated with this linear

action. Determining which subspaces of a particular Lie power of V are stabilised

by certain subgroups of GL(V ) is therefore a potential method of distinguishing

between these subgroups. We will explore this notion further later in this thesis.

Now, recall that if k > 2 is an integer, and if V is a vector space over a field of

characteristic 0 or characteristic greater than k, then we can identify AkV with a

subspace of T (V ).

Lemma 2.11.8 ([7, §3]). If char(F) 6= 2, then L2V is an irreducible F[GL(V )]-

module isomorphic to A2V . If char(F) /∈ 2, 3, then A3V is a subspace of A2V ⊗V ,

and L3V is an irreducible F[GL(V )]-module isomorphic to (A2V ⊗ V )/A3V .

We can easily calculate the dimensions of A2V and (A2V ⊗V )/A3V using Lemma

2.7.3, and this yields the following.

Corollary 2.11.9 ([7, §2]). If char(F) 6= 2, then dim(L2V ) =(d2

)= (d2 − d)/2. If

char(F) /∈ 2, 3, then dim(L3V ) = (d3 − d)/3.

Chapter 3

Induced linear groups and

universal p-groups

3.1 Inducing a linear group on P/Φ(P )

Throughout this chapter, let p be a prime, let d > 1 be an integer, and let V :=

Fdp. If P is a p-group, then since Φ(P ) is a characteristic subgroup of P , it follows

from Proposition 2.1.2 that there is a natural action θ : Aut(P ) → Aut(P/Φ(P )).

We will write A(P ) to denote the group induced by Aut(P ) on P/Φ(P ), i.e., the

image of θ. If P has rank d, then Burnside’s Basis Theorem implies that the Frattini

quotient P/Φ(P ) is the vector space V , and hence Aut(P/Φ(P )) ∼= Aut(V ) =

GL(V ). We can therefore identify A(P ) with a linear group H 6 GL(V ) ∼= GL(d, p)

such that the linear action of A(P ) on P/Φ(P ) is equivalent to that of the linear

action of H on V , in the sense that V and P/Φ(P ) are isomorphic as H-modules.

If Q is a p-group isomorphic to P , then Q/Φ(Q) and P/Φ(P ) are isomorphic H-

modules, and A(Q) and A(P ) are the images of equivalent representations. In

particular, A(Q) and A(P ) are conjugate in GL(d, p).

Recall from Section 2.3 that the lower central series of P is a descending series of

characteristic subgroups γi(P ) of P . Additionally, the lower exponent-p central series

of P , defined in Definition 2.4.8, is a descending series of characteristic subgroups

Pi(P ) of P . The nilpotency class of P is the smallest integer i such that γi+1(P ) = 1,

and the exponent-p class of P is the smallest integer i such that Pi+1(P ) = 1.

Observe that if P is elementary abelian, then P/Φ(P ) ∼= P . In this case, the

natural action θ is the identity map, and A(P ) = Aut(P ) ∼= GL(d, p). Therefore, in

order to induce a proper subgroup of GL(d, p) on a p-group of rank d, this p-group

must either have nilpotency class at least 2 or exponent at least p2, i.e., exponent-

p class at least 2. We can therefore consider this as a non-abelian1 analogue of

representing an arbitrary group as a linear group that acts on a vector space, i.e.,

an elementary abelian p-group, via standard representation theory. Later in this

chapter, we will explore how appropriate p-groups can be constructed as quotients

1More precisely, this is an analogue for p-groups that are not elementary abelian. However, wewill see that all p-groups that are relevant to our specific problem, introduced in Chapter 1 andsummarised below, are non-abelian.

41

42 Chapter 3. Induced linear groups and universal p-groups

of universal groups with respect to certain group theoretic properties, and how we

can determine the induced linear group in each case.

Recall from Chapter 1 that the main purpose of this thesis is to construct

“small” p-groups related to the (finite) simply connected version G of each excep-

tional Chevalley group G defined over Fp. We will define these groups in Chapter

4. In order to precisely describe what we mean by a “small” p-group, we introduce

the following definition.

Definition 3.1.1. Let H be a subgroup of GL(d, p), and let PH be the set of p-

groups P of rank d such that A(P ) = H. We say that a p-group P ∈PH is optimal

with respect to H if:

(i) no group in PH has a smaller exponent-p class than P ;

(ii) no group in PH with the same exponent-p class as P has a smaller exponent

than P ;

(iii) no group in PH with the same exponent-p class and exponent as P has a

smaller nilpotency class than P ; and

(iv) no group in PH with the same exponent-p class, nilpotency class and exponent

as P has a smaller order than P .

Indeed, for each G ∈ G2(p), F4(p), E8(p), we construct a p-group that is opti-

mal with respect to the normaliser of G in GL(d, p), where d is the minimal dimension

of a nontrivial irreducible Fp[G]-module. Furthermore, for each G ∈ G2(p), E8(p),we construct a p-group that is optimal with respect to G itself. In the remain-

ing cases, where G ∈ E6(p), E7(p), we construct a p-group that, with respect to

NGL(d,p)(G), satisfies conditions (i)–(iii) of Definition 3.1.1 and a stronger version of

condition (iv) that we specify in the next section.

3.2 Groups of prime exponent

In this section, we discuss a method of constructing p-groups of prime exponent

and nilpotency class 2 or 3, and then consider the linear group induced on the

Frattini quotient of such a p-group. Let B be the free Burnside group B(d, p) of

rank d and exponent p, and for a positive integer r, let Γ(d, p, r) := B/γr+1(B).

Lemma 3.2.1 ([7, §2]). The group Γ(d, p, r) is a p-group of rank d, exponent p and

nilpotency class at most r. Additionally, each p-group of rank d, exponent p and

nilpotency class r is a quotient of Γ(d, p, r).

Note that the nilpotency class of Γ(d, p, r) may indeed be less than r. For

example, each group of exponent 2 is abelian, and hence Γ(d, 2, 2) has nilpotency

class 1. If the nilpotency class of Γ(d, p, r) is equal to r, then we call Γ(d, p, r)

3.2. Groups of prime exponent 43

the universal p-group of rank d, exponent p and nilpotency class r, as it is the

unique (up to isomorphism) largest p-group with these properties. Indeed, we can

consider Γ(d, p, r) as the “free” group with these properties. It is noteworthy that

these universal p-groups are always finite, as they are defined as quotients of free

Burnside groups, which are often infinite.

Observe that Γ(d, p, 1) is the unique (up to isomorphism) elementary abelian

p-group of rank d, which we can identify with the vector space V . Recall that the

first Lie power L1V of V is V itself. The following theorem of Bamberg et al. [7,

§2] describes how Γ(d, p, 2) and Γ(d, p, 3) can be constructed using Lie powers of V .

Here, and throughout this section, we use without reference the properties of Lie

brackets given by Definition 2.11.1 and Proposition 2.11.2. By Proposition 2.11.4,

these apply to the bracket operation [·, ·] : L(V )→ L(V ) defined in Definition 2.11.3.

Theorem 3.2.2.

(i) Let Γ2(V ) be the set V × L2V equipped with the multiplication defined by

(a, b)(f, g) := (a+ f, b+ g + [a, f ]) (3.2.1)

for (a, b), (f, g) ∈ Γ2(V ). When p > 2, Γ2(V ) is a group of nilpotency class 2

and order pd(d+1)/2, and Γ2(V ) ∼= Γ(d, p, 2).

(ii) Let Γ3(V ) be the set V ×L2V ×L3V equipped with the multiplication defined

by

(a, b, c)(f, g, h) := (a+ f, b+ g + [a, f ], c+ h+ 3([b, f ]− [g, a]) + [a, f, f − a])

(3.2.2)

for (a, b, c), (f, g, h) ∈ Γ3(V ). When p > 3, Γ3(V ) is a group of nilpotency

class 3 and order pd(d+1)(2d+1)/6, and Γ3(V ) ∼= Γ(d, p, 3).

Hence there exists a universal p-group of rank d, exponent p and nilpotency

class r when r = 2 and p > 2, and when r = 3 and p > 3. Observe that the

products of group elements here are well-defined, since if a, f ∈ V and b, g ∈ L2V ,

then [a, f ] ∈ L2V and [b, f ], [g, a], [a, f, f − a] ∈ L3V by Proposition 2.11.6. Ad-

ditionally, if (a, b) ∈ Γ2(V ) and (a, b, c) ∈ Γ3(V ), then (a, b)−1 = (−a,−b) and

(a, b, c)−1 = (−a,−b,−c), while the identity of each group is the element with 0 in

each coordinate. The orders of the groups Γ2(V ) and Γ3(V ) follow from Corollary

2.11.9. Note that Bamberg et al. also show that when p > 3, Γ(d, p, 4) is isomorphic

to a group of nilpotency class 4 with underlying set V ×L2V ×L3V ×L4V . However,

we will not consider this group in this thesis.

It is clear that, when p > 2, R := (0, b) | b ∈ L2V is a subgroup of Γ2(V )

isomorphic to the elementary abelian group L2V . If f ∈ V and b, g ∈ L2V , then

(0, b)(f,g) = (−f,−g)(0, b)(f, g) = (−f, b− g + [−f, 0])(f, g) = (−f, b− g)(f, g)

= (0, b+ [−f, f ]) = (0, b).


Hence each subgroup of R is central in L2V . In fact, since L2V is a vector space over

a field of prime order, where scalar multiplication is equivalent to repeated vector

addition, the subgroups and subspaces of L2V coincide exactly. Hence if U is a

proper subspace of L2V , then U is a proper subgroup of R. It is easy to see that we

can identify PU := Γ2(V )/U with the set

V × (L2V )/U, (3.2.3)

equipped with the multiplication given by

(a, b+ U)(f, g + U) := (a+ f, b+ g + [a, f ] + U) (3.2.4)

for a, f ∈ V and b, g ∈ L2V . Here, (a, b + U)−1 = (−a,−b + U). Note also that

P0 = Γ2(V ).

Proposition 3.2.3. Suppose that p > 2, and let U be a proper subspace of L2V .

Then:

(i) PU′ = Φ(PU) = (0, b+ U) | b ∈ L2V , which is isomorphic to (L2V )/U ;

(ii) PU/Φ(PU) ∼= V ; and

(iii) PU is an extension of (L2V )/U by V .

Proof. Let (a, b + U), (f, g + U) ∈ PU , and let R := (0, b + U) | b ∈ L2V , which

is a subgroup of PU that is isomorphic to the elementary abelian group (L2V )/U .

We have

[(a, b+ U), (f, g + U)] = (−a,−b+ U)(−f,−g + U)(a, b+ U)(f, g + U)

= (−a− f,−b− g + [−a,−f ] + U)(a+ f, b+ g + [a, f ] + U)

= (0, [−a,−f ] + [a, f ] + [−a− f, a+ f ] + U)

= (0, [a, f ] + [a, f ]− [a, f ]− [f, a] + U) = (0, 2[a, f ] + U),

which is an element of R. Hence PU′ 6 R. Now, let e1, . . . , ed be a basis for V ,

let i, j ∈ 1, . . . , d, and let m := (p+ 1)/2. Then

[(ei, U), (ej, U)]m = (0, 2[ei, ej] + U)m = (0, 2m[ei, ej] + U),

and this is equal to (0, [ei, ej] + U), since 2m ≡ 1 (mod p). Proposition 2.11.6 and

the bilinearity of [·, ·] imply that L2V is spanned by the Lie brackets [ei, ej]. This

means that (L2V )/U is spanned by the vectors [ei, ej]+U , and hence R is generated

by the commutators [(ei, U), (ej, U)]. Therefore, PU′ = R.

As Γ2(V ) has exponent p, its nontrivial quotient PU also has exponent p. Hence

Φ(PU) = PU′ by Corollary 2.4.5. We have shown that U < Φ(P0) = Φ(Γ2(V )),

and so Proposition 2.4.7 implies that PU has the same rank as Γ2(V ), which is

d by Lemma 3.2.1. Hence PU/Φ(PU) ∼= V by Burnside’s Basis Theorem. Since

Φ(PU) ∼= (L2V )/U , it follows that PU is an extension of (L2V )/U by V .


Note that the above proposition follows easily from [7, §2] in the case PU = Γ2(V ).

Lemma 3.2.4. Suppose that p > 2, and let P be a p-group. Then P has rank d,

exponent p and nilpotency class 2 if and only if P ∼= PU for some proper subspace

U of L2V .

Proof. Let N be a normal subgroup of Γ2(V ). As Γ2(V ) has nilpotency class 2,

γ3(Γ2(V )) = 1. Therefore, Proposition 2.3.3 implies that Γ2(V )/N has nilpotency

class 2 if and only if N does not contain γ2(Γ2(V )) = (Γ2(V ))′. Moreover, Γ2(V ) has

rank d by Lemma 3.2.1, and so Proposition 2.4.7 implies that Γ2(V )/N has rank d

if and only if N 6 Φ(Γ2(V )). Furthermore, Γ2(V ) has exponent p, and so Γ2(V )/N

has exponent p if and only if N < Γ2(V ). We therefore have from Proposition 3.2.3

that Γ2(V )/N has rank d, exponent p and nilpotency class 2 if and only if N is a

proper subgroup of (0, b) | b ∈ L2V , i.e., a proper subspace of L2V .

It now follows from the definition of PU that if P ∼= PU for some proper subspace

U of L2V , then P has nilpotency class 2, rank d and exponent p. Conversely, if

P is a p-group of rank d, exponent p and nilpotency class 2, then P is a quotient

of Γ2(V ) by Lemma 3.2.1 and Theorem 3.2.2. Hence we require P ∼= PU for some

proper subspace U of L2V .

Now, suppose that p > 3, and let R be the subset (0, 0, c) | c ∈ L3V of Γ3(V ).

It is clear that this is a subgroup of Γ3(V ) isomorphic to the elementary abelian

group L3V . If f ∈ V , g ∈ L2V and c, h ∈ L3V , then

(0, 0, c)(f,g,h) = (−f,−g,−h)(0, 0, c)(f, g, h)

= (−f,−g + [−f, 0], c− h+ 3([−g, 0]− [0,−f ]) + [−f, 0, f ])(f, g, h)

= (−f,−g, c− h)(f, g, h)

= (0, [−f, f ], c+ 3([−g, f ]− [g,−f ]) + [−f, f, 2f ]) = (0, 0, c).

Hence each subgroup of R is central in L3V . As in the case of L2V and Γ2(V )

above, the proper subspaces of L3V are the proper subgroups of R. If W is such a

subspace, then we can identify QW := Γ3(V )/W with the set

V × L2V × (L3V )/W, (3.2.5)

equipped with the multiplication given by

(a, b, c+W )(f, g, h+W ) := (a+f, b+g+[a, f ], c+h+3([b, f ]−[g, a])+[a, f, f−a]+W )

(3.2.6)

for a, f ∈ V , b, g ∈ L2V and c, h ∈ L3W . Here, (a, b, c + W )−1 = (−a,−b, c + W ).

Note also that Q0 = Γ3(V ).

Proposition 3.2.5. Suppose that p > 3, and let W be a proper subspace of L3V .

Then:


(i) γ3(QW ) = (0, 0, c+W ) | c ∈ L3V , which is isomorphic to (L3V )/W ;

(ii) QW′ = Φ(QW ) = (0, b, c+W ) | b ∈ L2V, c ∈ L3V ;

(iii) QW/Φ(QW ) ∼= V ; and

(iv) QW′/γ3(QW ) ∼= L2V .

Proof. Let ai ∈ V , bi ∈ L2V and ci ∈ L3V for each i ∈ 1, 2, 3. Bamberg et al. [7,

p. 2936] show that, in Γ3(V ), [(a1, b1, c1), (a2, b2, c2), (a3, b3, c3)] = (0, 0, 12[a1, a2, a3]).

By comparing (3.2.2) and (3.2.6), we see that

[(a1, b1, c1 +W ), (a2, b2, c2 +W ), (a3, b3, c3 +W )] = (0, 0, 12[a1, a2, a3] +W ).

The group γ3(QW ) = [QW , QW , QW ] is generated by commutators of three elements

of QW (see [12, Proposition 1.6.5]), and so it lies in R := (0, 0, c+W ) | c ∈ L3V ,which is a subgroup of QW isomorphic to the elementary abelian group (L3V )/W .

Now, let e1, . . . , ed be a basis for V . For each i, j, k ∈ 1, . . . , d, we have

[(ei, 0,W ), (ej, 0,W ), (ek, 0,W )] = (0, 0, 12[ei, ej, ek +W ]). As p > 3, there is a posi-

tive integer m < p such that 12m ≡ 1 (mod p), and thus (0, 0, 12[ei, ej, ek] +W )m =

(0, 0, [ei, ej, ek] +W ). Proposition 2.11.6 and the bilinearity of [·, ·] imply that L3V

is spanned by the Lie brackets [ei, ej, ek], and it follows that R is generated by the

commutators [(ei, 0,W ), (ej, 0,W ), (ek, 0,W )]. Hence R = γ3(QW ).

Next, observe by comparing (3.2.1) and (3.2.6) that the first two coordinates of

[(a1, b1, c1+W ), (a2, b2, c2+W )] are equal to the two coordinates of [(a1, b1), (a2, b2)] ∈Γ2(V ). Thus by Proposition 3.2.3, QW

′ ⊆ S := (0, b, c+W ) | b ∈ L2V, c ∈ L3V ,which is easily seen to be a subgroup of QW . It also follows from this proposition

that for each b ∈ L2V , we can multiply commutators of elements of Γ3(V ) to obtain

an element (0, b, h+W ) of QW , for some h ∈ L3V . Since QW′ contains γ3(QW ), we

have shown above that, for each c ∈ L3V , the element (0, 0, c−h+W ) is a product of

commutators of elements of Γ3(V ). As (0, b, h+W )(0, 0, c−h+W ) = (0, b, c+W ), the

subgroup S is generated by commutators of elements of Γ3(V ). Therefore, QW′ = S.

Since Γ3(V ) has exponent p, the nontrivial quotient QW of Γ3(V ) also has ex-

ponent p. Hence Φ(QW ) = QW′ by Corollary 2.4.5. We have shown that W <

Φ(P0) = Φ(Γ3(V )), and so Proposition 2.4.7 implies that QW has the same rank

as Γ3(V ), which is d by Lemma 3.2.1. In addition, it is easy to see that the map

from QW′/γ3(QW ) to L2V defined by γ3(QW )(0, b, c + W ) 7→ b for b ∈ L2V and

c ∈ L3V is an isomorphism, and hence QW′/γ3(QW ) ∼= L2V .

The above proposition follows easily from [7, §2] in the case QW = Γ3(V ).

Lemma 3.2.6. Suppose that p > 3, and let P be a p-group. Then P has rank d,

exponent p and nilpotency class 3 if and only if P is a quotient of Γ3(V ) by a normal

subgroup that lies in γ2(Γ3(V )), and that does not contain γ3(Γ3(V )). In particular,


if P ∼= QW for some proper subspace W of L3V , then P has rank d, exponent p and

nilpotency class 3.

Proof. Let N be a normal subgroup of Γ3(V ). Lemma 3.2.1 and Theorem 3.2.2

imply that if P is a p-group of rank d, exponent p and nilpotency class 3, then P is

a quotient of Γ3(V ). Furthermore, if P ∼= QW for some proper subspace W of L3V ,

then W is a proper subgroup of (0, 0, c) | c ∈ L3V , which is equal to γ3(Γ3(V ))

by Proposition 3.2.5. Since γ2(Γ3(V )) contains γ3(Γ3(V )), it suffices to prove that

Γ3(V )/N has rank d, exponent p and nilpotency class 3 if and only if N has the

specified properties.

Since Γ3(V ) has nilpotency class 3, γ4(Γ3(V )) = 1. Hence Proposition 2.3.3 gives

that Γ3(V )/N has nilpotency class 3 if and only if N does not contain γ3(Γ3(V )).

Moreover, Γ3(V ) has rank d, and it follows from Proposition 2.4.7 that Γ3(V )/N

has rank d if and only if N 6 Φ(Γ3(V )), which is equal to (Γ3(V ))′ = γ2(Γ3(V ))

by Proposition 3.2.5. In addition, Γ3(V ) has exponent p, and thus Γ3(V )/N has

exponent p if and only if N < Γ3(V ). Therefore, Γ3(V )/N has rank d, exponent p

and nilpotency class 3 if and only if N is as specified.

Note that if N is a normal subgroup of Γ3(V ) that lies in γ2(Γ3(V )) and neither

contains nor lies in γ3(Γ3(V )), then Γ3(V )/N is a p-group of rank d, exponent p and

nilpotency class 3 that cannot be expressed as QW for any subspace W of L3V .

Recall from Proposition 2.11.7 that GL(d, p) acts linearly on L2V and on L3V .

Lemma 3.2.7 ([7, Theorem 2.5]). The group GL(d, p) acts on Γ2(V ), with (a, b)α :=

(aα, bα) for all a ∈ V , b ∈ L2V and α ∈ GL(d, p). Similarly, GL(d, p) acts on Γ3(V ),

with (a, b, c)α := (aα, bα, cα) for all a ∈ V , b ∈ L2V , c ∈ L3V and α ∈ GL(d, p).

As above, let B be the free Burnside group B(d, p). If r is a positive integer,

then B/γr+1(B) = Γ(d, p, r) is a finite p-group by Lemma 3.2.1, and it contains the

subgroup γr(B)/γr+1(B) by the Correspondence Theorem.

Lemma 3.2.8 ([7, §2]). Let r be a positive integer, and suppose that γr(B) 6= 1.

Then γr(B)/γr+1(B) is an Fp[GL(d, p)]-module isomorphic to LrV .

It follows from Lemma 3.2.8 and the Correspondence Theorem that the subspaces

of LrV can be identified with the quotients M/γr+1(B), for the subgroups M of

γr(B) that contain γr+1(B). Observe that [M,B] 6 [γr(B), B] = γr+1(B) 6M , and

hence M C6 B. This means that M/γr+1(B) C6 B/γr+1(B), and the finite p-group

Γ(d, p, r)/(M/γr+1(B)) = (B/γr+1(B))/(M/γr+1(B)) is isomorphic to B/M by the

Third Isomorphism Theorem. The next result describes the linear group induced

on the Frattini quotient of this p-group.

Theorem 3.2.9 ([7, Theorem 2.2]). Let r be a positive integer, and let M be a

proper subgroup of γr(B) that contains γr+1(B). Then A(B/M) is the stabiliser of

M/γr+1(B) in GL(d, p).


The following theorem, which describes A(P ) when P is a p-group of exponent

p and nilpotency class 2, or when P is isomorphic to QW for some proper subspace

W of L3V , is a more detailed version of a result used in the proof [7, §6] of Theorem

1.0.3. Recall from Lemma 3.2.4 that a p-group has exponent p and nilpotency class

2 if and only if it is isomorphic to PU for some proper subspace U of L2V .

Theorem 3.2.10. Suppose that P has exponent p.

(i) Suppose that p > 2, and that P has nilpotency class 2, i.e., that P ∼= PU for

some proper subspace U of L2V . Then A(P ) = GL(d, p)U .

(ii) Suppose that p > 3, and that P ∼= QW for some proper subspace W of L3V .

Then A(P ) = GL(d, p)W .

Proof. Let r ∈ 2, 3, and suppose that p > r. Additionally, let X be the proper

subspace of LrV such that P is isomorphic to PX or QX . As in the discussion below

Lemma 3.2.8, we have X = M/γr+1(B), where M is a proper subgroup of γr(B)

that contains γr+1(B), and P ∼= B/M . It follows from Theorem 3.2.9 that, up to

conjugacy in GL(d, p), A(P ) = A(B/M) = GL(d, p)X .

This theorem shows that in order to construct a p-group P as PU (respectively,

as QW ) such that A(P ) is a particular subgroup H of GL(d, p), then H must be the

stabiliser in GL(d, p) of some proper subspace of L2V (respectively, of L3V ). We

must therefore be able to distinguish between H and any proper overgroup of H in

GL(d, p) by comparing the subspaces of L2V (or of L3V ) stabilised by these linear

groups. Lemma 2.11.8 shows that GL(d, p) acts irreducibly on L2V (and on L3V ).

This means that if H is a maximal subgroup of GL(d, p), as in Theorem 1.0.3, then

it is sufficient to observe that H acts reducibly on L2V (or on L3V ). Otherwise, we

must consider how the proper overgroups of H act on the Lie power.

Observe that if r ∈ 2, 3, if p > r, and if P = Γr(V ) ∈ P0, Q0, then

A(P ) = GL(d, p)0 = GL(d, p). Hence if X is a proper subspace of LrV , then the

linear group induced on the Frattini quotient of Γr(V )/X is a subgroup of the linear

group induced on the Frattini quotient of Γr(V ).

Suppose now that p > 3, that H is a particular subgroup of GL(d, p), and that

there is no p-group P of exponent-p class 2 such that A(P ) = H. Additionally,

suppose that W is a proper subspace of L3V such that A(QW ) = H, with W having

the largest dimension of such a proper subspace. Then QW has minimal order among

the groups that can be expressed as QX , with X a proper subspace of L3V such that

A(QX) = H, and QW has exponent-p class 3 by Proposition 2.4.9. It is also clear

that if Q is an optimal p-group with respect to H, as in Definition 3.1.1, then Q has

exponent-p class 3, exponent p and nilpotency class 3. However, we are not able to

determine the order of Q. This is because, if N is a normal subgroup of Γ3(V ) that

satisfies the hypotheses of Lemma 3.2.6 but does not lie in γ3(Γ3(V )) ∼= L3V , then

3.3. p-covering groups and p-groups of exponent-p class 2 49

Theorem 3.2.10(ii) does not yield any information about A(Γ3(V )/N). We therefore

introduce the following definition.

Definition 3.2.11. Let H be a subgroup of GL(d, p), and suppose that each p-group

that is optimal with respect to H has exponent-p class 3, exponent p and nilpotency

class 3. Additionally, let QH be the set of p-groups QW , for proper subspaces W of

L3V , such that A(QW ) = H. Then a p-group P ∈ QH is quasi-optimal with respect

to H if no group in QH has a smaller order than P .

Observe that a p-group that is quasi-optimal with respect to H satisfies condi-

tions (i)–(iii) of Definition 3.1.1 and a stronger version of condition (iv).

3.3 p-covering groups and p-groups of exponent-p class 2

We now discuss the concept of the p-covering group of a p-group. We will use

general results about p-covering groups in order to show how the p-covering group

of the elementary abelian group of rank d can be used to construct each p-group of

rank d and exponent-p class 2, and in order to determine the linear group induced

on the Frattini quotient of such a p-group.

Throughout this section, P denotes a p-group of rank d > 1. Recall from Defini-

tion 2.1.10 that, for a group G and an integer k, Gk is the subgroup of G generated

by the k-th powers of all elements of G. We will also write QP to denote the set

of p-groups Q of rank d that contain an elementary abelian subgroup KQ such that

KQ 6 Z(Q) ∩ Φ(Q) and Q/KQ∼= P . Note that each subgroup of KQ is a normal

subgroup of Q, as KQ 6 Z(Q).

The following theorem is from a paper by O’Brien [62, Theorem 2.2]. Our for-

mulation of this theorem includes information that can be derived from O’Brien’s

paper, and that is explicitly stated in a subsequent paper by Eick, Leedham-Green

and O’Brien [21, p. 2274].

Theorem 3.3.1. There exists a unique (up to isomorphism) group P ∗, called the

p-covering group of P , such that P ∗ ∈ QP and such that each group in QP is a

quotient of P ∗. If P has exponent-p class c, then the exponent-p class of P ∗ is at

most c+ 1.

The subgroup KP ∗ of P ∗ is called the p-multiplicator of P [62, p. 679–680].

Theorem 3.3.2 ([21, p. 2275]). Let ψ be the natural epimorphism from P ∗ to P with

kernel KP ∗ . Additionally, let α ∈ Aut(P ). Then α lifts via ψ to an automorphism

α∗ of P ∗ that stabilises KP ∗ . Furthermore, Aut(P ) acts on KP ∗ , with kα := kα∗

for

each k ∈ KP ∗ and each α ∈ Aut(P ).

Theorem 3.3.3 ([62, Theorem 2.4]). Suppose that P has exponent-p class c, and

let X be a subgroup of KP ∗ . Then the following are equivalent.


(i) Q := P ∗/X has rank d and exponent-p class c+ 1, and Q/Pc+1(Q) ∼= P .

(ii) X < KP ∗ and XPc+1(P ∗) = KP ∗ .

Since Pc+1(Q) is a characteristic subgroup of Q, Proposition 2.1.2 implies that

Aut(Q) acts naturally on Q/Pc+1(Q) ∼= P . The next result follows from [62, Theo-

rem 2.10] (see also [21, Theorem 3.2]).

Theorem 3.3.4. Suppose that P has exponent-p class c, let X be a subgroup of

KP ∗ that satisfies the equivalent conditions of Theorem 3.3.3, and let Q := P ∗/X.

Additionally, let θ : Aut(Q) → Aut(Q/Pc+1(Q)) ∼= Aut(P ) be the natural action.

Then (Aut(Q))θ is the stabiliser of X in Aut(P ), with respect to the action of

Aut(P ) on KP ∗ .

We now apply the above results to elementary abelian groups and groups of

exponent-p class 2. Let E be the elementary abelian p-group of rank d, which has

exponent-p class 1, and which we can identify with the vector space V . As E∗ ∈ QE,

E∗ has rank d. Observe that, for an arbitrary p-group P , we have P2(P ) = P ′P p,

which is equal to Φ(P ) by Proposition 2.4.4. The following two results are our own.

Proposition 3.3.5. The p-multiplicator KE∗ of E is Φ(E∗), and E∗ has exponent-

p class 2. Furthermore, if X is a proper subgroup of Φ(E∗), then it satisfies the

equivalent conditions of Theorem 3.3.3, and E∗/X has rank d and exponent-p class

2. Conversely, every p-group of rank d and exponent-p class 2 is a quotient of E∗ by

a proper subgroup of Φ(E∗).

Proof. The quotient E∗/KE∗ is isomorphic to the elementary abelian group E,

and hence Φ(E∗) 6 KE∗ by Theorem 2.4.3. Additionally, KE∗ lies in Φ(E∗) by

definition, and hence KE∗ = Φ(E∗). Next, let X be a proper subgroup of KE∗ .

Then XP2(E∗) = XΦ(E∗) = XKE∗ = KE∗ , and hence X satisfies the equivalent

conditions of Theorem 3.3.3. Since E has exponent-p class 1, this theorem implies

that E∗/X has rank d and exponent-p class 2.

Next, let P be a p-group of rank d and exponent-p class 2. Definition 2.4.8 gives

[Φ(P ), P ] = Φ(P )p = 1, i.e., Φ(P ) is elementary abelian and lies in Z(P ). Moreover,

it follows from Burnside’s Basis Theorem that P/Φ(P ) ∼= E. Hence P ∈ QE, and so

Theorem 3.3.1 implies that P ∼= E∗/Y for some normal subgroup Y of E∗. As E∗

and P both have rank d, Proposition 2.4.7 implies that Y 6 Φ(E∗). In fact, since

P is not elementary abelian, we have Y < Φ(E∗) by Theorem 2.4.3.

Finally, the direct product of d copies of the cyclic group Cp2 is abelian of ex-

ponent p2, and thus it has exponent-p class 2 by Proposition 2.4.9. This group also

has rank d, and it is therefore a quotient of E∗. Thus E∗ is not elementary abelian.

As the exponent-p class of E∗ is at most 2 by Theorem 3.3.1, it follows that E∗ has

exponent-p class 2.


Thus E∗ is the universal p-group of rank d and exponent-p class 2.

Proposition 3.3.6. Suppose that p > 2. Then (E∗)p < Φ(E∗), and the group

Γ(d, p, 2) is isomorphic to E∗/(E∗)p.

Proof. First, Γ(d, p, 2) is the largest group of rank d, exponent p and nilpotency

class 2 by Lemma 3.2.1 and Theorem 3.2.2. It follows from Proposition 2.4.9 that

Γ(d, p, 2) is also the largest group of rank d, exponent p and exponent-p class 2. Thus

Proposition 3.3.5 implies that if X is a proper subgroup of Φ(E∗) of the smallest

possible order such that E∗/X has exponent p, then E∗/X ∼= Γ(d, p, 2). We therefore

have from Lemma 2.1.12 that X = (E∗)p, and hence (E∗)p < Φ(E∗).

Now, let F be the free group of rank d. Although F is not a p-group, we

can define Pi(F ) for each positive integer i as in Definition 2.4.8. If P is a p-

group of rank d, with P ∼= F/R for some normal subgroup R of F , then P ∗ ∼=F/([R,F ]Rp) [62, p. 679]. We have E ∼= F/P2(F ) [14, p. 416], and hence E∗ ∼=F/([P2(F ), F ](P2(F ))p) = F/P3(F ). Furthermore, Φ(F/P3(F )) = P2(F )/P3(F )

[14, p. 416], and Proposition 3.3.5 implies that this group is isomorphic to the p-

multiplicator Φ(E∗) of E.

The p-multiplicator of E∗ is elementary abelian by definition, and so can be

identified with a vector space over Fp. With V = Fdp, the natural linear action of

Aut(E) ∼= GL(V ) on E ∼= V induces an action of GL(V ) on Φ(E∗) by Theorem

3.3.2. Indeed, when p > 2, Φ(E∗) ∼= P2(F )/P3(F ) is a faithful Fp[GL(V )]-module

isomorphic to V ⊕ L2V [14, p. 420–421]. Here, the action of GL(V ) on V is the

natural linear action, the linear action of GL(V ) on L2V is defined as in Definition

2.11.7, and the action of GL(V ) on V ⊕ L2V is coordinatewise. This means that

Φ(E∗) ∼= V × L2V as an elementary abelian group. As E∗/Φ(E∗) ∼= V , it follows

that E∗ is an extension of V ×L2V by V . We summarise this in the following lemma.

Lemma 3.3.7. Suppose that p > 2. Then E∗ is an extension of V ×L2V by V , and

the p-multiplicator Φ(E∗) of E is an Fp[GL(V )]-module isomorphic to V ⊕ L2V .

Thus the subgroups of Φ(E∗) are the subspaces of V ⊕L2V . Since A2V ∼= L2V as

Fp[GL(V )]-modules when p > 2 by Lemma 2.11.8, a version of Lemma 3.3.7 appears

in a paper by Glasby, Palfy and Schneider [29, §1–2]. Our next result identifies the

GL(V )-submodules of V ⊕L2V with particular subgroups of Φ(E∗). Versions of this

result and of the subsequent theorem also appear in the aforementioned paper.

Lemma 3.3.8. Suppose that p > 2. Then GL(V ) acts linearly on the groups (E∗)p

and (E∗)′. Moreover, (E∗)p ∼= V and (E∗)′ ∼= L2V as Fp[GL(V )]-modules, and the

Fp[GL(V )]-module Φ(E∗) is equal to the direct sum (E∗)p ⊕ (E∗)′ of submodules.

Proof. For each α ∈ Aut(E), the automorphism α∗ ∈ Aut(E∗) defined in Theorem

3.3.2 stabilises the characteristic subgroups (E∗)′ and (E∗)p of E∗. Furthermore,


(E∗)′(E∗)p = Φ(E∗) by Proposition 2.4.4, and this is equal to KE∗ by Proposition

3.3.5. Thus the action of Aut(E) ∼= GL(V ) on Φ(E∗) defined in Theorem 3.3.2

induces an action of GL(V ) on each of (E∗)′ and (E∗)p. Lemma 3.3.7 implies that

the action of GL(V ) on Φ(E∗) is linear, with Φ(E∗) ∼= V ⊕ L2V as Fp[GL(V )]-

modules, and hence each of (E∗)′ and (E∗)p is a GL(V )-submodule of V ⊕ L2V .

Recall from Lemma 2.11.8 that L2V is an irreducible Fp[GL(V )]-module isomorphic

to A2V . The Fp[GL(V )]-module V is also irreducible, and V 6∼= L2V by Proposition

2.8.24. It follows from Corollary 2.8.16 that V ⊕L2V has exactly two nonzero proper

submodules, namely, the direct summands V and L2V .

Now, we have Γ(d, p, 2) ∼= E∗/(E∗)p from Proposition 3.3.6. Moreover, Propo-

sition 3.2.3 implies that Γ(d, p, 2) is an extension of L2V by V , while Lemma 3.3.7

implies that E∗ is an extension of (V × L2V ) by V . Hence (E∗)p is nontrivial, and

it is a proper subgroup of Φ(E∗) by Proposition 3.3.6. Thus (E∗)p is isomorphic

to either V or L2V as a GL(V )-module. We have V ∼= E∗/(V × L2V ), which is

isomorphic to (E∗/V )/((V × L2V )/V ) by the Third Isomorphism Theorem, and

this in turn is isomorphic to (E∗/V )/L2V . Hence E∗/V is an extension of L2V

by V . Similarly, E∗/L2V is an extension of V by V . As the action of GL(V ) on

Γ(d, p, 2) ∼= E∗/(E∗)p involves the action of GL(V ) on L2V by Lemma 3.2.7, it

follows that (E∗)p ∼= V as a GL(V )-module.

Finally, since Γ(d, p, 2) ∼= E∗/(E∗)p has nilpotency class 2 by Theorem 3.2.2,

Corollary 2.1.7 implies that (E∗)′ is not a subgroup of (E∗)p. In particular, (E∗)′

is nontrivial. Furthermore, the direct product of d copies of the cyclic group Cp2 is

abelian of rank d and exponent p2. This group has exponent-p class 2 by Proposition

2.4.9, and so Proposition 3.3.5 implies that it is a quotient of E∗ by a proper subgroup

of Φ(E∗). We also have from Corollary 2.1.7 that this proper subgroup contains

(E∗)′. Thus (E∗)′ is a nontrivial proper subgroup of Φ(E∗) distinct from (E∗)p, and

hence (E∗)′ ∼= L2V as a GL(V )-module.

The following result is a generalisation of Theorem 3.2.10(i). In order to state

this theorem, we use the fact that a p-group has rank d and exponent-p class 2 if

and only if it is the quotient of E∗ by a proper subgroup of Φ(E∗), as stated in

Proposition 3.3.5. Recall also the definition of PU from Section 3.2.

Theorem 3.3.9. Suppose that p > 2, and that P has exponent-p class 2, i.e., that

P ∼= E∗/X for a proper subgroup X of Φ(E∗) ∼= V ⊕L2V . Then A(P ) = GL(d, p)X .

Furthermore:

(i) P is abelian if and only if X contains the direct summand L2V . Here, P has

exponent p2; and A(P ) = GL(d, p)X∩V , with X ∩ V a proper subspace of V .

(ii) P has exponent p if and only if X contains the direct summand V . Here, P

has nilpotency class 2; A(P ) = GL(d, p)U , where U is the proper subspace

X ∩ L2V of L2V ; and P ∼= PU .


Proof. Let Q := E∗/X. Proposition 3.3.5 implies that X satisfies the equivalent

conditions of Theorem 3.3.3. Since E has exponent-p class 1, it follows from Theorem

3.3.4 that the image of the natural action θ : Aut(Q) → Aut(Q/P2(Q)) is the

stabiliser of X in Aut(E) ∼= GL(d, p). We have P2(Q) = Φ(Q), and hence the image

of θ is A(Q). As P ∼= Q, it follows that A(P ) = A(Q) = GL(d, p)X , up to conjugacy

in GL(d, p).

Now, since P has exponent-p class 2, we have from Proposition 2.4.9 that if P

is abelian, then it has exponent p2, and if instead P has exponent p, then it has

nilpotency class 2. Furthermore, P is abelian if and only if (E∗)′ 6 X by Corollary

2.1.7, in which case L2V ⊆ X by Lemma 3.3.8. Here, A(P ) is the stabiliser in

GL(d, p) of X = (X ∩ V ) ⊕ L2V . Since the full group GL(d, p) stabilises L2V by

Proposition 2.11.7, A(P ) is the stabiliser of X ∩ V . As X is a proper subspace of

V ⊕L2V that contains the direct summand L2V , X ∩ V is a proper subspace of V .

Similarly, Lemma 2.1.12 shows that P has exponent p if and only if (E∗)p 6 X,

in which case V ⊆ X by Lemma 3.3.8. Here, A(P ) is the stabiliser in GL(d, p) of

X = V ⊕ (X ∩L2V ). Since GL(d, p) stabilises V , A(P ) is the stabiliser of X ∩L2V .

In this case, the Third Isomorphism Theorem gives E∗/X ∼= (E∗/(E∗)p)/(X/(E∗)p).

We can identify X/(E∗)p with the quotient (V ⊕ (X∩L2V ))/V , which is isomorphic

to X ∩ L2V = U , and this is a proper subspace of L2V . Additionally, E∗/(E∗)p ∼=Γ(d, p, 2) by Proposition 3.3.6. Hence P ∼= E∗/X ∼= Γ(d, p, 2)/U , which is isomorphic

to Γ2(V )/U = PU by Theorem 3.2.2.

We can draw a conclusion from the above theorem similar to the remark following

Theorem 3.2.10. Namely, in order to construct a p-group P of exponent-p class 2

such that A(P ) is a particular subgroupH of GL(d, p), we must be able to distinguish

between H and any proper overgroup of H in GL(d, p) by comparing the subspaces

of V ⊕ L2V stabilised by these linear groups.

Observe that if P is a p-group satisfying condition (i) of Theorem 3.3.9, with

X∩V 6= 0, then A(P ) acts reducibly on V . In fact, A(P ) is a maximal C1-subgroup

of GL(d, p) [7, Remark 6.3]. If instead P = E∗, then Theorem 3.3.9 implies that

A(P ) = GL(d, p)0 = GL(d, p).

The final theorem in this chapter reveals more information about a given p-group

of exponent-p class 2.

Theorem 3.3.10 ([29, Theorem 4]). Suppose that p > 2, and that P and X are as

in Theorem 3.3.9. Then P has a unique proper nontrivial characteristic subgroup if

and only if GL(d, p)X acts irreducibly on each of V and (V ⊕ L2V )/X.

A p-group with a unique (proper nontrivial) characteristic subgroup is called

a UCS p-group [29]. Since each group in the lower exponent-p central series of a

p-group is characteristic, each UCS p-group has exponent-p class at most 2.

Chapter 4

Simple groups of Lie type and

highest weight theory

4.1 The simple groups of Lie type

Throughout this chapter, we will use highest weight theory to explore the struc-

tures of Lie powers of minimal modules for groups related to simple groups of Lie

type, which are particular finite non-abelian simple groups. In this section, we briefly

introduce these simple groups using information given by Kleidman and Liebeck [53,

Ch. 2.9, Ch. 5.1].

Using Lie notation, the simple groups of Lie type are denoted tY`(q), where

t ∈ 1, 2, 3, Y ∈ A,B,C,D,E, F,G, ` is a positive integer, and q is a prime

power. Only certain combinations of t, Y , ` and q are allowed, and we will specify

these shortly. For certain combinations of t and Y , tY`(q) is a classical group, and

otherwise it is an exceptional group of Lie type. Furthermore, if t ∈ 2, 3, then the

group is twisted . If instead t = 1, then Y`(q) := 1Y`(q) is a Chevalley group [13,

p. 153]. Finally, we say that the group tY`(q) is defined over Fq.Tables 4.1.1 and 4.1.2 list the (simple) classical groups of Lie type and the (sim-

ple) exceptional groups of Lie type, respectively, specifying the allowed combinations

of t, Y , ` and q. The former table also gives the standard classical group notation

for each group, as defined in Section 2.6. When ` > 3, there are no isomorphisms

between groups in these tables, but when ` 6 3, some of the classical groups of Lie

type are isomorphic to each other or to simple alternating groups. Additionally,

since Ω(2`+ 1, q) is non-abelian and simple whenever q is odd and (`, q) 6= (1, 3) by

Proposition 2.6.9, this group has trivial centre and is therefore equal to PΩ(2`+1, q).

Note that the simple groups 2B2(q) are often called Suzuki groups , and the simple

groups 2G2(q) and 2F4(q) are often called Ree groups [73, p. 134, p. 163].

Table 4.1.3 lists the order of each group of Lie type and its Schur multiplier, i.e.,

the order of the centre of its universal cover.

The Tits group 2F4(2)′

is another non-abelian simple group that is sometimes

considered as an exceptional group of Lie type. However, we will not consider this

group as a simple group of Lie type. Later in this thesis, we will define G2(q), with

q odd, and F4(q), with q a power of a prime p > 3, as automorphism groups of

55

56 Chapter 4. Simple groups of Lie type and highest weight theory

Table 4.1.1: The classical groups of Lie type.

Lie notation Classical group notation

A`(q), (`, q) /∈ (1, 2), (1, 3) PSL(`+ 1, q)2A`(q), (`, q) /∈ (1, 2), (1, 3), (2, 2) PSU(l + `, q)

B`(q), q odd, (`, q) 6= (1, 3) Ω(2`+ 1, q)

C`(q), (`, q) /∈ (1, 2), (1, 3), (2, 2) PSp(2`, q)

D`(q), ` > 2 PΩ+(2`, q)2D`(q), ` > 1 PΩ−(2`, q)

Table 4.1.2: The exceptional groups of Lie type.

Lie notation

G2(q), q > 22B2(22m+1), m a positive integer2G2(32m+1), m a positive integer

F4(q)2F4(22m+1), m a positive integer

3D4(q)

E6(q)2E6(q)

E7(q)

E8(q)

certain algebras. Furthermore, Wilson [73, Ch. 4] provides definitions of all of the

exceptional groups of Lie type.

4.2 Highest weight theory

Let G be a simple group of Lie type defined over the finite field Fq of charac-

teristic p. In this section, we introduce highest weight theory, which will allow us

to investigate the structures of modules for an infinite group related to G. Later in

this chapter, we will see that we can then derive information about related modules

for a certain finite group, which in some cases is G. Throughout this section, and

in a later section in this chapter, we will illustrate definitions and results using the

example of A1(q) = PSL(2, q), with q > 3.

Lemma 4.2.1 ([56, p. 135]). Let J be the universal cover of G, let P be the Sylow

p-subgroup of Z(J), and let G := J/P . Then G is isomorphic to G/Z(G), via an

isomorphism that preserves the group’s action on each G-module over each extension

field of Fq. Furthermore, G is the set of fixed points of a particular endomorphism

of a certain linear algebraic group G defined over the algebraic closure of Fp.

4.2. Highest weight theory 57

Table 4.1.3: The orders of the groups of Lie type and their Schur multipliers.

Group Group order Schur multiplier order

A`(q)1

(`+1,q−1)q`(`+1)/2

∏`+1i=2(qi − 1)

2, if (`, q) ∈ (1, 4), (2, 2), (3, 2),6, if (`, q) = (1, 9),

48, if (`, q) = (2, 4),

(`+ 1, q − 1), otherwise

2A`(q)1

(`+1,q+1)q`(`+1)/2

∏`+1i=2(qi − (−1)i)

2, if (`, q) ∈ (1, 4), (3, 2),6, if (`, q) = (1, 9),

12, if (`, q) = (5, 2),

36, if (`, q) = (3, 3),

(`+ 1, q + 1), otherwise

B`(q) 12q`

2∏ì=1(q2i − 1)

6, if (`, q) ∈ (1, 9), (3, 3),2, otherwise

C`(q)1

(2,q−1)q`

2∏ì=1(q2i − 1)

2, if (`, q) ∈ (1, 4), (3, 2),6, if (`, q) = (1, 9),

(2, q − 1), otherwise

D`(q)1

(4,q`−1)q`(`−1)(q` − 1)

∏`−1i=1(q2i − 1)

2, if (`, q) = (3, 2),

4, if (`, q) = (4, 2),

(4, q` − 1), otherwise

2D`(q)1

(4,q`+1)q`(`−1)(q` + 1)

∏`−1i=1(q2i − 1)

2, if (`, q) ∈ (2, 2), (3, 2),6, if (`, q) = (2, 3),

36, if (`, q) = (3, 3),

(4, q` + 1), otherwise

G2(q) q6(q6 − 1)(q2 − 1)

2, if q = 4,

3, if q = 3,

1, otherwise

2B2(q) q2(q2 + 1)(q − 1)4, if q = 8,

1, otherwise2G2(q) q3(q3 + 1)(q − 1) 1

F4(q) q24(q12 − 1)(q8 − 1)(q6 − 1)(q2 − 1)2, if q = 2,

1, otherwise2F4(q) q12(q6 + 1)(q4 − 1)(q3 + 1)(q − 1) 13D4(q) q12(q8 + q4 + 1)(q6 − 1)(q2 − 1) 1

E6(q) 1(3,q−1)

q36∏

i∈2,5,6,8,9,12(qi − 1) (3, q − 1)

2E6(q) 1(3,q+1)

q36∏

i∈2,5,6,8,9,12(qi − (−1)i)

12, if q = 2,

(3, q + 1), otherwise

E7(q) 1(2,q−1)

q63∏

i∈2,6,8,10,12,14,18(qi − 1) (2, q − 1)

E8(q) q120∏

i∈2,8,12,14,18,20,24,30(qi − 1) 1

We call G the (finite) simply connected version of G [13, p. 268]. The linear

algebraic group G is an infinite group that can be considered as a group of matrices

with entries in Fp [13, p. 268]. Malle and Testerman [60, Ch. 1.1] give a precise

definition of a linear algebraic group, and of a morphism of linear algebraic groups,

which is a particular type of group homomorphism. In fact, the endomorphism in


Lemma 4.2.1 is an endomorphism of linear algebraic groups. Note also that Fp = Fq.Throughout the rest of this chapter, we use the following notation:

G is a simple group of Lie type defined over the finite field Fq of characteristic p;

G is the simply connected version of G;

K := Fp; and

G is the linear algebraic group defined over K such that G is the set of fixed

points of an endomorphism of G.

Table 4.1.3 shows that for all but a finite number of groups G, the Sylow p-

subgroup of the Schur multiplier of G is trivial, and thus G is the universal cover

of G. In particular, this is the case if G ∈ E6(q), E7(q). We obtain the following

using the definition of the universal cover of G and Lemma 4.2.1.

Corollary 4.2.2. Let J be the universal cover of G. If Z(J) is a p-group, then

G ∼= G. In particular, this is the case if Z(J) = 1, i.e., if J ∼= G.

The next result is a consequence of Corollary 4.2.2 and Table 4.1.3.

Corollary 4.2.3. Let G be an exceptional group of Lie type. Then G ∼= G, unless

G = E6(q) with q ≡ 1 (mod 3); G = 2E6(q) with q ≡ 2 (mod 3); or G = E7(q) with

q odd.

Theorem 4.2.4 (Tits [60, Theorem 24.17]). The group G is quasisimple.

Proof. The universal cover J of G is perfect by Lemma 2.2.5. It follows from

Proposition 2.1.6 that, for the Sylow p-subgroup P of Z(J), we have

(J/P )′ = (PJ/P )′ = PJ ′/P = PJ/P = J/P.

Hence J/P is perfect, and this group is equal to G by Lemma 4.2.1. We also have

from this lemma that G/Z(G) is isomorphic to the non-abelian simple group G, and

thus G is quasisimple.

Proposition 4.2.5 ([60, p. 193]). Suppose that G = tY`(q), and let H = sY`(r) be a

simple group of Lie type, with s ∈ 1, 2, 3, and with r a power of p. Then the linear

algebraic group G associated with G is also the linear algebraic group associated

with H. In particular, if G 6= H, then G and H are the sets of fixed points of two

distinct endomorphisms of G.

In other words, the linear algebraic group associated with tY`(pf ), where f is an

integer, does not depend on t or f . We therefore follow the notation of Malle and

Testerman [60, p. 142] and write G = Y` for the linear algebraic group corresponding

to G = tY`(q), where it is implicit that G is defined over K = Fp. In this chapter:

A` denotes a linear algebraic group and not an alternating group.


Malle and Testerman [60, Ch. 3, Ch. 6] present the following definitions and

results related to the linear algebraic group G, some of which do not apply to all

linear algebraic groups. Note that the general linear group of any vector space

over K, including the multiplicative group K× ∼= GL(1, K) of K, can be given

the structure of a linear algebraic group, as can any finite direct product of linear

algebraic groups [60, p. 3–4].

Definition 4.2.6. A character of G is a morphism of linear algebraic groups (and

hence a group homomorphism) from G to K×.

Proposition 4.2.7. The set X(G) of characters of G is an abelian group called

the character group of G. In particular, if we write the group operation of X(G)

additively, then (x)(f1 + f2) := (x)f1(x)f2 for all x ∈ G and all f1, f2 ∈ X(G).

For the following definition, an isomorphism of linear algebraic groups is a bi-

jective morphism of linear algebraic groups whose inverse is also such a morphism

[60, p. 20–21]. It is therefore also a group isomorphism.

Definition 4.2.8. Let n be a nonnegative integer. A subgroup of G is called a torus

of dimension n if it is isomorphic, as a linear algebraic group, to the direct product

of n copies of K×.

Proposition 4.2.9. The linear algebraic group G contains a maximal torus, i.e., a

torus T 6 G such that there is no torus T ′ satisfying T < T ′ 6 G. All maximal tori

of G have the same dimension, which is called the rank of G.

In this chapter, when we refer to the rank of a linear algebraic group, we mean

the dimension of its maximal tori, and not the minimum size of a generating set for

the group. Note that the rank of the linear algebraic group Y` is ` [8, p. 229].

Proposition 4.2.10. Let T be an n-dimensional torus. Then the character group

X(T ) is isomorphic to Zn. In particular, if T is a maximal torus of a linear algebraic

group of rank `, then X(T ) ∼= Z`.

We now fix a maximal torus T of the linear algebraic group G.

The following definition involves the Lie algebra g of G over K and the adjoint

representation Ad : G→ GL(g) of G, which Malle and Testerman [60, Ch. 7] de-

fine precisely. In this thesis, all K-representations of G are rational representations,

which are representations ρ : G→ GL(V ), for some vector space V over K, with ρ

a morphism of linear algebraic groups [60, p. 32]. Similarly, all K[G]-modules are

rational modules, i.e., modules that afford rational representations. Note that each

submodule of a rational K[G]-module is itself rational, as are modules constructed

via tensor products, exterior powers and quotients (of submodules constructed sim-

ilarly) of rational K[G]-modules (see [60, Ch. 5.2, Ch. 15–16]).


Definition 4.2.11 ([60, Ch. 8.1]). For α ∈ X(T ), we define

gα := v ∈ g | v(t)Ad = (t)αv for all t ∈ T.

The root system of G (with respect to T ) is the set

Φ := α ∈ X(T ) | α 6= 1X(T ), gα 6= 0,

where 1X(T ) is the trivial character mapping each t ∈ T to 1 ∈ K×. The elements

of Φ are called the roots of G.

Example 4.2.12. When G = PSL(2, q) for q > 3, we have G = SL(2, q) and G =

A1 = SL(2, K) [60, p. 193, p. 237]. Furthermore, rank(G) = 1, and the subgroup

T :=

(a 0

0 a−1

)| a ∈ K×

of SL(2, K) is isomorphic, as a linear algebraic group,

to K× [60, p. 38]. Thus T is a maximal torus of G by Proposition 4.2.9. Since T has

dimension 1, X(T ) = t 7→ ait | i ∈ Z [60, p. 23]. Additionally, g is the Lie algebra

of 2 × 2 matrices of trace 0 with entries in K, with [v, w] = vw − wv for v, w ∈ g

[60, p. 54], where multiplication here is standard matrix multiplication. For t ∈ Tand v ∈ g, we have v(t)Ad = t−1vt [60, p. 52].

We now calculate the corresponding root system Φ of G. Suppose that a non-

trivial character α ∈ X(T ) is such that gα 6= 0, and let 0 6= v :=

(w x

y −w

)∈ gα,

with w, x, y ∈ K. Then by Definition 4.2.11, for each t =

(at 0

0 a−1t

)∈ T , we have

(t)α

(w x

y −w

)= (t)αv = t−1vt =

(w a−2

t x

a2ty −w

).

Since α is nontrivial, we require w = 0. If x 6= 0, then (t)α = a−2t for all t ∈ T .

Here, we must have y = 0, since there exists at ∈ K× with a2t 6= a−2

t , and thus

gα =

(0 x

0 0

)| x ∈ K. Similarly, if x = 0 and y 6= 0, then (t)α = a2

t for all t ∈ T ,

and gα =

(0 0

y 0

)| y ∈ K. We have therefore shown that Φ = αa, αb, where

(t)αa := a2t and (t)αb := a−2

t for all t ∈ T .

Now, X(T ) ∼= Z` by Proposition 4.2.10, and since Z` ⊂ R`, it follows that we

can consider X(T ) and Φ ⊂ X(T ) as subsets of R`, with 1X(T ) = 0 [60, Ch. 9.1]. For

example, when T has dimension 1, the character t 7→ ait corresponds to i ∈ Z ⊂ R.

Thus as a subset of R, the root system of G in Example 4.2.12 is Φ = −2, 2.

Definition 4.2.13 ([60, p. 63–64]). The Weyl group of Φ is W := 〈sα | α ∈ Φ〉,where each sα ∈ GL(`,R) is the reflection in the hyperplane orthogonal to α.


Proposition 4.2.14 ([60, p. 63–64]). The group W preserves a unique (up to mul-

tiplication by a scalar) symmetric bilinear form (·, ·) on R` such that (α, α) > 0 for

all nonzero α ∈ R`. Moreover, the action of W on R` induces an action on X(T ).

This proposition implies that if α, β ∈ R` have different lengths with respect to

(·, ·), i.e., if (α, α) 6= (β, β), then there is no element w ∈ W such that αw = β. The

following result strengthens this conclusion when α, β ∈ Φ.

Proposition 4.2.15 ([60, Corollary A.18]). The set of root lengths (α, α) | α ∈ Φcontains at most two elements. Furthermore, for α, β ∈ Φ, we have (α, α) = (β, β)

if and only if there exists an element w ∈ W such that αw = β.

Definition 4.2.16 ([60, p. 276]). Suppose that the roots of G have two different

lengths with respect to (·, ·). Then the roots with the smaller length are the short

roots of G, and the roots with the greater length are the long roots of G.

Proposition 4.2.17 ([60, p. 64]). There exists a subset ∆ := α1, . . . , α` of Φ

such that ∆ is a vector space basis for R` and such that, for all α ∈ Φ, we have

α =∑`

i=1 ciαi, with either ci ∈ Z>0 for all i, or ci ∈ Z60 for all i. Such a subset is

called a base of Φ.

We will henceforth consider a fixed ordered base ∆ = α1, . . . , α` of Φ.

Proposition 4.2.18 ([56, p. 136]). Let 6 be the relation on X(T ) such that α 6 β

for α, β ∈ X(T ) if and only if β − α =∑`

i=1 ciαi, with ci ∈ R>0 for all i. Then 6 is

a partial order.

We will also write α < β when α 6 β and α 6= β.

Definition 4.2.19 ([60, p. 126–127]). The fundamental dominant weights of T (with

respect to ∆) are the elements of the unique set λ1, . . . , λ` of characters in X(T )

such that

2(λi, αj)

(αj, αj)=

1, if i = j

0, if i 6= j.

Example 4.2.20. We have shown that if G = PSL(2, q), then Φ = −2, 2. Since

2 is a basis for R, we can choose ∆ = 2, i.e., α1 = 2. Now, s2 ∈ W is the

reflection that maps 2 to −2, i.e., that maps u to −u for each u ∈ R. This is the

same reflection as s−2, and hence W = 1W , s2. The usual inner product on Rgiven by 〈u, v〉 = uv for u, v ∈ R is a symmetric bilinear form with 〈u, u〉 > 0 for all

nonzero u, and we have 〈us2 , vs2〉 = 〈−u,−v〉 = (−u)(−v) = uv = 〈u, v〉. Thus the

bilinear form (·, ·) from Proposition 4.2.14 is given by (u, v) := µuv for all u, v ∈ R,

where µ ∈ R is some nonzero scalar. The fundamental dominant weight λ1 is the

unique scalar such that 1 = 2 (λ1,α1)(α1,α1)

= 2 (λ1,2)(2,2)

= 2µλ1·2µ·2·2 , i.e., λ1 = 1.


Proposition 4.2.21 ([56, p. 136]). Each character inX(T ) can be expressed uniquely

as a Z-linear combination of the fundamental dominant weights of T .

Definition 4.2.22 ([56, p. 136]). We say that a character λ ∈ X(T ) is dominant if

λ =∑`

i=1 ciλi, with ci ∈ Z>0 for all i.

Definition 4.2.23 ([60, p. 122]). Let V be a (finite-dimensional) K[G]-module. For

λ ∈ X(T ), we define

Vλ := v ∈ V | vt = (t)λv for all t ∈ T.

If Vλ 6= 0, then λ is a weight of V with (respect to T ) with multiplicity dim(Vλ),

and Vλ is a weight space of V .

Observe that T acts linearly on V , as it is a subgroup of G, and that Vλ is a

subspace of V . Since (t)λ ∈ K× for all t ∈ T , it follows that Vλ is a submodule of the

K[T ]-module V . We will write Λ(V ) to denote the weight multiset for V , with each

weight represented as many times as its multiplicity. This means that the elements

of Λ(V ) can be associated with distinct 1-dimensional subspaces of V . Note that

there may exist distinct elements λ, µ ∈ Λ(V ) such that λ = µ as elements of X(T ).

Note that if U is a K[G]-module isomorphic to V , then Λ(U) = Λ(V ) [36, p. 144].

Malle and Testerman [60, Lemma 15.3] prove the following, using the fact that

W ∼= NG(T )/T .

Lemma 4.2.24. Let V be a K[G]-module with weight λ. Then for each element w

of the Weyl group W , λw is a weight of V with the same multiplicity as λ.

Lemma 4.2.25 ([60, p. 122]). The weights of the Lie algebra of G are 0, with

multiplicity equal to the rank of G, and the roots of G, each with multiplicity 1.

Proposition 4.2.26 ([60, p. 122]). Let V be aK[G]-module. Then V =⊕

λ∈X(T ) Vλ.

This proposition yields the following corollary.

Corollary 4.2.27. Let V be a K[G]-module, and let B(V ) be the union of (any

choice of) bases for the weight spaces of V . Then B(V ) is a basis for V , with a

1-1 correspondence between basis vectors and elements of Λ(V ), via the associated

1-dimensional subspaces of V . In particular, dim(V ) = |Λ(V )|.

Lemma 4.2.28 ([67, p. 106–108]). Let V and U be K[G]-modules. Then Λ(V ⊗U)

is the multiset of sums [λ + µ | λ ∈ Λ(V ), µ ∈ Λ(U)], with λ1 + µ1 = λ2 + µ2

if and only if λ1 = λ2 and µ1 = µ2. Additionally, if k is an integer such that

2 6 k 6 n := dim(V ), and if Λ(V ) = [λ1, λ2, . . . , λn], then

Λ(AkV ) = [λi1 + λi2 + · · ·+ λik | 1 6 i1 < i2 < · · · < ik 6 n].


Proof. Let v ∈ B(V ) and u ∈ B(U). From Corollary 4.2.27, there exist weights

λ ∈ Λ(V ) and µ ∈ Λ(U) such that v ∈ Vλ and u ∈ Uµ. Using Definition 4.2.23 and

the definition of the action of T on V ⊗ U , we have that, for all t ∈ T ,

(v ⊗ u)t = vt ⊗ ut = ((t)λv)⊗ ((t)µu) = (t)λ(t)µ(v ⊗ u) = (t)(λ+ µ)(v ⊗ u).

Thus the tensor v⊗ u is an element of (V ⊗U)λ+µ, and hence λ+ µ ∈ Λ(V ⊗U). If

there exist vectors v′ ∈ B(V ) ∩ Vλ and u′ ∈ B(U) ∩ Uµ such that v 6= v′ or u 6= u′,

then v⊗ u and v′⊗ u′ are distinct elements in the usual basis for U ⊗ V . These two

tensors therefore correspond to different elements of Λ(V ⊗ U), and it follows that

Λ(V⊗U) contains [λ+µ | λ ∈ Λ(V ), µ ∈ Λ(U)]. Since dim(V⊗U) = dim(V ) dim(U),

Corollary 4.2.27 implies that this is the full multiset Λ(V ⊗U). A similar proof that

uses Lemma 2.7.3 and the definition of the action of T on AkV shows that Λ(AkV )

is as required.

Lemma 4.2.29 ([36, Lemma 10.37]). Let V be a K[G]-module with submodule U .

Then Λ(V/U) = Λ(V ) \ Λ(U).

Proof ([36]). Let λ be a weight of V , and let π : V → V/U be the natural linear

projection map defined by (v)π := v + U for v ∈ V . Definition 4.2.23 and the

definition of the action of T on V/U imply that for each v ∈ Vλ and each t ∈ T ,

((v)π)t = (v + U)t = vt + U = (vt)π = ((t)λv)π = (t)λ(v)π,

and hence (Vλ)π ⊆ (V/U)λ. Additionally, the kernel of π|Vλ is Vλ ∩ U = Uλ. Thus

Λ(V/U) contains Λ(V ) \ Λ(U), and since dim(V/U) = dim(V )− dim(U), Corollary

4.2.27 implies that these multisets are equal.

Equivalently, Λ(V ) is the disjoint union of Λ(U) and Λ(V/U). Applying this fact

recursively to the submodules in a composition series for V yields the following.

Corollary 4.2.30. Let V be a K[G]-module. Then Λ(V ) is the disjoint union of

the weight multisets for the composition factors of V .

Proposition 4.2.31 ([13, Ch. 5.1.1–5.1.2]). Let H be either the linear algebraic

group G or its subgroup G, and let φ be the map from H to itself that transforms

each matrix in H by raising each of its entries to its p-th power. Then φ is an

automorphism of H. For each nonnegative integer i, the automorphism φi is called

a field automorphism of H. Moreover, the field automorphisms of G induce the field

automorphisms of G. If G is a Chevalley group with q = pe for a positive integer e,

then the set of distinct nontrivial field automorphisms of G is φ, φ2, . . . , φe−1.

Note that the nontrivial field automorphisms of G are group automorphisms,

but not automorphisms of linear algebraic groups [60, Remark 11.13]. Recall from

Proposition 2.8.8 that, given an automorphism α of G, we can twist a K[G]-module

V by α to obtain the K[G]-module V α. The following lemma generalises a claim

from the proof of [60, Proposition 16.6].


Lemma 4.2.32. Let k be a nonnegative integer, and let V be a K[G]-module. Then

Λ(V φk) = [pkλ | λ ∈ Λ(V )].

Proof. Let v ∈ B(V ), and let f and fφk be the maps associated with the K[G]-

modules V and V φk , respectively. From Corollary 4.2.27 and Definition 4.2.23,

there exists λ ∈ Λ(V ) such that, for all t ∈ T , (v, t)f = vt = (t)λv. We have

(v, t)fφk = (v, tφk)f , and tφ

k= tp

k ∈ T [60, Theorem 16.5]. Thus (v, t)fφk = (tpk)λv.

This is equal to ((t)λ)pkv, since λ is a homomorphism, and hence equal to (t)(pkλ)v

by Proposition 4.2.7. Therefore, pkλ ∈ Λ(V φk). Since B(V φk) = B(V ), Corollary

4.2.27 implies that [pkλ | λ ∈ Λ(V )] is the full multiset Λ(V φk).

Theorem 4.2.33 ([56, p. 136]). Let U be an irreducible K[G]-module. Then there

exists a unique weight λ of U , called the highest weight of U , such that µ 6 λ

for each weight µ of U , where 6 is the partial order defined in Proposition 4.2.18.

Moreover, U is uniquely determined (up to isomorphism) by its highest weight,

which is dominant and has multiplicity 1.

Theorem 4.2.34 ([56, p. 136]). Let λ ∈ X(T ) be dominant. Then there exists an

irreducible K[G]-module with highest weight λ.

We write L(λ) to denote “the” irreducible K[G]-module with highest weight λ.

For a general K[G]-module V , we say that a weight λ in a subset D of Λ(V ) is a

highest weight of D if µ 6 λ for all µ ∈ D. If D = Λ(V ), then we call λ a highest

weight of V .

Observe that if U is the trivial irreducible K[G]-module, then Definition 4.2.23

implies that the trivial character 0 ∈ R` is the only weight of U . Thus U = L(0)

and Λ(U) = [0]. We now work through a nontrivial example.

Example 4.2.35. The natural embedding of G = SL(2, K) in GL(2, K) is a mor-

phism of linear algebraic groups [60, p. 5, p. 7], and so the 2-dimensional vector

space U over K is a (rational) K[G]-module. For each nonzero vector u ∈ U , at

least one of A =

(1 1

0 1

)∈ G and the transpose AT ∈ G maps u to a vector in

U that is not a scalar multiple of u. Thus G does not stabilise a nontrivial proper

subspace of U , i.e., U is an irreducible K[G]-module.

Now, for λ ∈ X(T ), t =

(at 0

0 a−1t

)∈ T and u = (x, y) ∈ U , with x, y ∈ K,

we have ut = (atx, a−1t y) and (t)λu = (aitx, a

ity) for some i ∈ Z. Thus by Definition

4.2.23, for λ to be a weight of U , we require λ to be the character that maps t to

at, with Uλ = (x, 0) | x ∈ K, or that maps t to a−1t , with Uλ = (0, y) | y ∈ K.

Therefore, as elements of R, the weights of U are 1 = λ1 and −1 = −λ1, each

with multiplicity 1, i.e., Λ(U) = [−1, 1]. We have 1− (−1) = 2 = α1, from Example

4.2.20, and thus −1 6 1. Hence 1 is the highest weight of U , and therefore U = L(1).


Our next proposition combines results from Bray, Holt and Roney-Dougal [13,

p. 269, p. 306]; Kleidman and Liebeck [53, p. 191–192]; and Malle and Testerman

[60, p. 88–90, p. 215].

Proposition 4.2.36.

(i) Suppose that G ∈ A`, D`, E6, with ` > 1. Then G has an automorphism of

order 2 called a graph automorphism. If G = D4, then G has an additional

graph automorphism of order 3. If G is a Chevalley group, then each graph

automorphism of G induces an automorphism of G of the same order, also

called a graph automorphism.

(ii) Suppose that (G, p) ∈ (B2, 2), (G2, 3), (F4, 2). Then G has an automor-

phism, called a graph automorphism, whose square is the field automorphism

φ of G. If G is a Chevalley group, then the graph automorphism of G induces

an automorphism of G, also called a graph automorphism, whose square is the

field automorphism of G induced by φ.

(iii) In all cases in (i) and (ii), each graph automorphism γ of G induces a per-

mutation τ on the set of fundamental dominant weights of T . Moreover, if V

is an irreducible K[G]-module with highest weight∑`

i=1 ciλi, then V γ is an

irreducible K[G]-module with highest weight∑`

i=1 ciλ(i)τ .

The following result is our own.

Lemma 4.2.37. Let C be a (possibly empty) disjoint union of weight multisets for

composition factors of the K[G]-module V . Additionally, let λ ∈ D := Λ(V ) \ Cbe such that λ 6< µ for all µ ∈ D. Then L(λ) is a composition factor of V . In

particular, this is the case if D has a highest weight λ.

Proof. By Corollary 4.2.30, Λ(V ) is the disjoint union of the weight multisets for

the composition factors of V . The subset D of Λ(V ) is also a disjoint union of weight

multisets for composition factors of V , and hence λ ∈ Λ(U) for some composition

factor U of V . This composition factor is irreducible by definition, and hence The-

orem 4.2.33 implies that U = L(µ), for some µ ∈ D with λ 6 µ. By the definition

of λ, we therefore have µ = λ and U = L(λ).

We can apply this lemma recursively to a K[G]-module to increase the multiset

of known composition factors, corresponding to C, and decrease the multiset of

unknown composition factors, corresponding to D. We illustrate this in the following

example.

Example 4.2.38. Let U be the irreducible K[SL(2, K)]-module L(1). From Exam-

ple 4.2.35, Λ(U) = [−1, 1]. Applying Lemma 4.2.28 gives

Λ(U ⊗ U) = [−1 + (−1),−1 + 1, 1 + (−1), 1 + 1] = [−2, 0, 0, 2],


and

Λ(A2(U ⊗ U)) = [−2 + 0,−2 + 0,−2 + 2, 0 + 0, 0 + 2, 0 + 2] = [−2,−2, 0, 0, 2, 2].

It is easy to see that −2 6 2 and 0 6 2, and hence Lemma 4.2.37 implies that L(2)

is a composition factor of U⊗U and of A2(U⊗U). Since Φ = −2, 2, we have from

Lemma 4.2.25 that the weight multiset for the Lie algebra g of SL(2, K) is [−2, 0, 2].

When q is odd, g is an irreducible K[G]-module [60, Theorem 15.20], and hence

L(2) ∼= g by Theorem 4.2.33. Thus Λ(L(2)) = [−2, 0, 2], and so dim(L(2)) = 3 by

Corollary 4.2.27. Applying Lemma 4.2.37 recursively shows that the composition

factors of U ⊗U are L(2) and L(0), and that the composition factors of A2(U ⊗U)

are two copies of L(2). On the other hand, g is reducible when q is even [60, Theorem

15.20], and thus L(2) is a composition factor of g, with Λ(L(2)) ⊂ Λ(g). We have

2 ∈ Λ(L(2)) by the definition of L(2), and since the reflection s2 ∈ W maps 2 to

−2, Lemma 4.2.24 implies that −2 ∈ Λ(L(2)). Thus Λ(L(2)) = [−2, 2]. It follows

that the composition factors of U ⊗U are L(2) and two copies of L(0), and that the

composition factors of A2(U ⊗ U) are two copies of each of L(0) and L(2).

Definition 4.2.39 ([13, p. 269]). For a positive integer r, let X(r) be the set of r-

restricted characters in X(T ), i.e., characters that are linear combinations∑`

i=1 ciλi

of fundamental dominant weights of T , with 0 6 ci < r for all i.

Observe that each λ ∈ X(T ) can be uniquely expressed as µ0 + pµ1 + · · ·+ pnµn,

where n is a nonnegative integer and where µi ∈ X(p) for each i. In addition, as the

action of G on L(0) is trivial, twisting L(0) by an automorphism of G has no effect.

Theorem 4.2.40 (Steinberg’s Tensor Product Theorem [13, Theorem 5.1.2]). Let

k be a nonnegative integer, and let µi ∈ X(p) for each i ∈ 1, . . . , k. Then

L(µ0 + pµ1 + · · ·+ pkµk) ∼= L(µ0)⊗ L(µ1)φ ⊗ · · · ⊗ L(µk)φk ,

where φ is the field automorphism of G defined in Proposition 4.2.31.

Thus each irreducible K[G]-module can be constructed by twisting irreducible

modules with p-restricted highest weights, and then forming their tensor product.

For example, suppose that G = SL(2, K) and p = 3. Then L(5) = L(2 + 3 · 1) ∼=L(2)⊗ L(1)φ and L(18) = L(32 · 2) ∼= L(2)φ

2. Using Lemmas 4.2.28 and 4.2.32 and

Example 4.2.38, we obtain Λ(L(18)) = [−18, 0, 18] and

Λ(L(5)) = [λ+ µ | λ ∈ [−3, 3], µ ∈ [−2, 0, 2]] = [−5,−3,−1, 1, 3, 5].

4.3 Multiplicity free K[G]-modules

In this section, we show that many K[G]-modules of interest are multiplicity free.

Later, we will use this fact to prove that related K[G]-modules and Fq[G]-modules

4.3. Multiplicity free K[G]-modules 67

are multiplicity free. Note that many of the definitions and results from Section 4.2

apply when we replace K with the field C of complex numbers. For example, the

linear algebraic group Y` can be defined over C, and Theorems 4.2.33 and 4.2.34

apply in this case [60, p. 82, Theorem 15.17]. We can also consider the character

group X(T ) ⊂ R` to be independent of the field over which Y` is defined. However,

we will retain our notation of writing G for the group Y` defined over K. As such,

we use different notation in the statement of the following result.

Lemma 4.3.1 ([56, p. 137]). Let F be equal to C or the algebraic closure of a

finite field, let H = Y` be defined over F, and let λ be a dominant character of

a maximal torus of H. Then there exists an F[H]-module V (λ), called the Weyl

module corresponding to λ, such that:

(i) dim(V (λ)) does not depend on F;

(ii) the irreducible F[H]-module L(λ) is a quotient of V (λ);

(iii) if F = C, then V (λ) ∼= L(λ); and

(iv) there exists a prime r such that, if p > r and F = Fp, then V (λ) ∼= L(λ).

There may also exist primes p < r such that V (λ) ∼= L(λ) when F = Fp.

Recall that W denotes the Weyl group of the root system G, while ∆ denotes

the base of this root system.

Proposition 4.3.2 ([60, p. 92–93]). There exists a unique element w0 ∈ W , called

the longest element of W , such that ∆w0 = −∆.

Proposition 4.3.3 ([60, p. 125, p. 132]). Let λ ∈ X(T ) be dominant. Then −λw0

is dominant, and L(−λw0) ∼= (L(λ))∗.

Proposition 4.3.4 ([41, p. 24, p. 118]). Let λ, µ ∈ X(T ) be dominant, and let M

be a K[G]-module whose composition factors are exactly L(λ) and L(µ), with L(µ)

isomorphic to a submodule U of M . If L(λ) is not isomorphic to a submodule W of

M , with M = U ⊕W , then either:

(i) µ < λ, and M is isomorphic to a quotient of V (λ); or

(ii) λ < µ, and M is isomorphic to a submodule of (V (−µw0))∗.

Now, let S be a finite subset of X(T ), with each λ ∈ S dominant. It follows from

Lemma 4.3.1 that there exists a prime r such that, for all λ ∈ S, the dimension of

L(λ) and its weight multiset remain constant for all p > r. We can also construct

a new K[G]-module M using irreducible modules L(λ) with λ ∈ S, via tensor

products, exterior powers and quotients (of submodules constructed similarly). As

long as p > r, the dimension of M and its weight multiset will not vary with p, since


these depend only on the weight multisets for the modules L(λ) and the construction

operations, by results from Section 4.2. However, the composition factors of M may

vary. If we redefine S to include all dominant weights of M , then we deduce that

there exists a prime r such that, for all p > r, M has the same number of composition

factors, with each having a constant dimension and highest weight. In the following

theorem, which is our own, r denotes this prime.

Theorem 4.3.5. Let M be a K[G]-module constructed from irreducible K[G]-

modules via tensor products, exterior powers and quotients. Suppose that the com-

position factors of M are the same as those of the corresponding module defined over

Fr, in terms of dimensions and highest weights, and suppose that these composition

factors are pairwise non-isomorphic. Then M is multiplicity free.

Proof. Let 0 = M0 ⊂M1 ⊂ · · · ⊂Mn = M be a composition series for M . Then

Theorem 4.2.33 implies that, for each i ∈ 1, 2, . . . , n, Mi/Mi−1∼= Li := L(µi)

for some dominant µi ∈ X(T ). Since the dimension and highest weight of each

composition factor of M are equal to those of the corresponding module defined

over Fr, Lemma 4.3.1 implies that Li ∼= V (µi) for each i. Hence V (µi) is irreducible.

Similarly, it follows from Proposition 4.3.3 that dim(L(−µw0i )) = dim((L(µi))

∗) =

dim(Li) is the same when defined overK, or over Fq for any q > r. Hence V (−µw0i ) ∼=

L(−µw0i ) by Lemma 4.3.1. The duals of isomorphic modules are isomorphic to each

other by Proposition 2.8.6, and so (V (−µw0i ))∗ ∼= (L(−µw0

i ))∗, which is isomorphic

to (L∗i )∗ by Proposition 4.3.3. This is isomorphic to Li by Proposition 2.8.7, and

thus (V (−µw0i ))∗ is irreducible.

Now, for a given j ∈ 2, 3, . . . , n, we see using the Correspondence Theorem and

the Third Isomorphism Theorem that the composition factors of Wj := Mj/Mj−2

are (Mj−1/Mj−2)/(Mj−2/Mj−2) ∼= Mj−1/Mj−2∼= Lj−1 and Wj/(Mj−1/Mj−2) ∼=

Mj/Mj−1∼= Lj, with the former isomorphic to a maximal submodule of Wj. Since

V (µj) and (V (−µw0j−1))∗ are irreducible, Proposition 4.3.4 implies that Lj is also

isomorphic to a maximal submodule of Wj. It follows from the Correspondence

Theorem that Mj contains a maximal submodule U such that U/Mj−2∼= Lj. Thus

M has a composition series

0 = M0 ⊂M1 ⊂ · · ·Mj−2 ⊂ U ⊂Mj ⊂ · · · ⊂Mn = M.

As the composition factors of this series are the same as the original series by the

Jordan-Holder Theorem, we have Mj/U ∼= Lj−1. This process of finding a new

composition series can be performed a total of (j − 1) times in order to obtain a

composition series whose smallest nonzero module, which is an irreducible submod-

ule of M , is isomorphic to Lj. Since the composition factors of M are pairwise

non-isomorphic, M is multiplicity free by Proposition 2.8.17.

McNinch [61, Corollary 1.1.1] proved a more uniform result about (not neces-

sarily multiplicity free) semisimple modules: if G = Y`, then each K[G]-module of

4.4. The linear algebraic group G2 69

2α + βα + β

−α

−2α− β −α− β

α

3α + 2β

β

−3α− β

−3α− 2β

−β

3α + β

Figure 4.4.1: The roots of the linear algebraic group G = G2.

dimension at most `p is semisimple. However, later in this chapter, we will use

Theorem 4.3.5 to prove the semisimplicity of K[G]-modules whose dimensions are

higher than this upper bound.

4.4 The linear algebraic group G2

We will now use the theory developed in Sections 4.2–4.3 to determine the struc-

tures of particular K[G]-modules, where G is a linear algebraic group corresponding

to an exceptional group of Lie type G. For the most part, we will investigate the ex-

terior squares of the irreducible modules L(λi) for the fundamental dominant weights

λi of a maximal torus of G. When p > 2, each exterior square can be identified with

the second Lie power of the irreducible module, by Lemma 2.11.8. Later in this

chapter, we will use the results here to derive information about the structures of

corresponding G-modules.

Proposition 4.4.1 ([60, Example 9.5, Example 15.21, p. 274–275]). The linear

algebraic group G = G2 has rank 2. We can choose ∆ = α, β, where α = (a, 0) ∈R2 for some a ∈ R, and β =

√3a(− cos π

6, sin π

6) ∈ R2. Then the root system Φ of G

is the union of the set of short roots Φs := ±α,±(α + β),±(2α + β) and the set

of long roots Φl := ±β,±(3α+ β),±(3α+ 2β). Furthermore, the Weyl group W

of Φ is isomorphic to the dihedral group D12. Finally, the fundamental dominant

weights of the corresponding maximal torus are λ1 = 2α + β and λ2 = 3α + 2β.

As illustrated in Figure 4.4.1, the short roots of G can be thought of as the

vertices of a regular hexagon in the plane R2 centred at the origin, with the long

roots corresponding to twice the midpoints of the hexagon’s edges (with respect to

the origin) [73, p. 103]. Note that λ1 and λ2 are the highest short root and highest

long root, respectively, with respect to the partial order 6, and λ1 < λ2. In addition,

there exist irreducible K[G]-modules L(λ1) and L(λ2) by Theorem 4.2.34.


We now determine the dimensions of the irreducible K[G]-modules L(λ1) and

L(λ2), and their weight multisets, for each p. In some cases, we obtain the dimension

of each module directly from a paper by Lubeck [56]. In fact, the dimension of each

module and its weight multiset can be derived using online data [58] that supple-

ments the results of the paper. However, this data was obtained computationally,

while we use a theoretical approach. Note that, for each linear algebraic group G

related to a simple group of Lie type, we label the fundamental dominant weights

using the same numbering as Malle and Testerman [60, Table 9.1]. However, in the

case G = G2, the weight λ1 in our notation is denoted λ2 by Lubeck, and vice versa.

Lemma 4.4.2. The irreducible K[G]-module L(λ1) has dimension 6 and weight

multiset Φs when p = 2, and dimension 7 and weight multiset Φs ∪ [0] otherwise.

Proof. Lubeck [56, Appendix A.49] states that the dimension of L(λ1) is as required

in each case. We have λ1 ∈ L(λ1) by definition, and λ1 ∈ Φs by Proposition 4.4.1.

Since Φs is an orbit of W by Proposition 4.2.15, Λ(L(λ1)) contains Φs by Lemma

4.2.24. Since |Φs| = 6, Corollary 4.2.27 implies that Λ(L(λ1)) = Φs when p = 2, and

that Λ(L(λ1)) contains exactly one element λ with λ /∈ Φs otherwise. Since each

nonzero vector in R2 is mapped to a different vector by at least one of the reflections

sα, sβ ∈ W , Lemma 4.2.24 implies that λ = (0, 0) = 0.

Lemma 4.4.3. The irreducible K[G]-module L(λ2) has dimension 7 and weight

multiset Φl ∪ [0] when p = 3, and dimension 14 and weight multiset Φ ∪ [0, 0]

otherwise. When p 6= 3, L(λ2) is the Lie algebra of G.

Proof. First, since λ2 is the highest root of G by Proposition 4.4.1, it follows from

Lemmas 4.2.25 and 4.2.37 that L(λ2) is a composition factor of the Lie algebra g

of G. In fact, g is irreducible when p 6= 3 [60, Theorem 15.20]. Thus in this case

L(λ2) is the Lie algebra of G, which has weight multiset Φ∪ [0, 0] by Lemma 4.2.25.

The dimension of L(λ2) follows from Corollary 4.2.27 and Proposition 4.2.15. Now,

when p = 3, Λ(L(λ2)) contains λ2 by definition. Propositions 4.2.15 and 4.4.1 imply

that Φl is an orbit of W containing λ2, and thus Λ(L(λ2)) contains Φl by Lemma

4.2.24. The dimension of L(λ2) here is 7 [56, Appendix A.49], and since |Φl| = 6,

Λ(L(λ2)) contains exactly one element λ with λ /∈ Φl. Only the zero vector is fixed

by W , and therefore λ = 0 by Lemma 4.2.24.

Throughout the rest of this chapter, we will often denote characters in X(T ) by

(c1, c2, . . . , cl) :=∑`

i=1 ciαi, with ∆ = α1, . . . , α` and each ci ∈ R. When each

ci is a nonnegative, single-digit integer, we will write (c1c2 · · · c`). Similar notation

is used by Lubeck [56], but with characters expressed as a linear combination of

fundamental dominant weights, instead of a linear combination of roots in ∆.

The next two results are our own.

4.4. The linear algebraic group G2 71

Theorem 4.4.4. Let V := L(λ1). For each value of p, the composition factors of

A2V and their dimensions are listed in Table 4.4.1. Furthermore, Λ(A2V ) is the

disjoint union of Φ and [0, 0, 0] if p = 2, and the disjoint union of Φ, Φs and [0, 0, 0]

otherwise. If p > 3, then A2V is multiplicity free.

Table 4.4.1: The composition factors of A2V and their dimensions and multiplicities,with G = G2 and V = L(λ1).

Condition on p Composition factor Dimension Multiplicity

p = 2L(λ2) 14 1L(0) 1 1

p = 3L(λ2) 7 1L(λ1) 7 2

p > 3L(λ2) 14 1L(λ1) 7 1

Proof. First, suppose that p = 2. It follows from Lemmas 4.2.28 and 4.4.2 that

Λ(A2V ) = [(10)− (10),±((10)± (11)),±((10)± (21)), (11)− (11),

± ((11)± (21)), (21)− (21)]

= [0,±(21),±(01),±(31),±(11), 0,±(32),±(10), 0]

= Φ ∪ [0, 0, 0],

where all ± symbols are independent. The highest weight of A2V is (32) = λ2, and

hence L(λ2) is a composition factor of A2V by Lemma 4.2.37. Lemma 4.4.3 implies

that Λ(A2V ) \ Λ(L(λ2)) = [0], and dimL(λ2) = 14. Applying Lemma 4.2.37 again

shows that the remaining composition factor of A2V is L(0), with dimension 1.

Next, suppose that p > 2. Compared to the case with p = 2, Λ(V ) has an

extra weight of 0 with multiplicity 1 by Lemma 4.4.2. Therefore, Lemma 4.2.28

implies that the additional elements of Λ(A2V ) are those obtained by adding 0 to the

elements of Φs. These are exactly the elements of Φs, and thus Λ(A2V ) is the disjoint

union of Φ, Φs and [0, 0, 0]. The highest weight of A2V is again (32) = λ2, which

means that L(λ2) is a composition factor of A2V by Lemma 4.2.37. If p = 3, then

we have from Lemma 4.4.3 that dim(L(λ2)) = 7, and that D := Λ(A2V ) \Λ(L(λ2))

is the disjoint union of [0, 0] and two copies of Φs. On the other hand, if p > 3,

then dim(L(λ2)) = 14, and D is the disjoint union of [0] and Φs. In either case,

(21) = λ1 is a highest weight of D, and hence Lemmas 4.2.37 and 4.4.2 give the

required composition factors and their dimensions. In particular, note that when

p = 3, λ1 is the highest weight of Λ(A2V ) \ C, where C is the disjoint union of

Λ(L(λ1)) and Λ(L(λ2)). Finally, since A2V has the same composition factors for

all p > 3, in terms of dimensions and highest weights, and since these are pairwise

non-isomorphic, Theorem 4.3.5 implies that A2V is multiplicity free when p > 3.


In the table mentioned in the following theorem and the proof of the theorem,

φ is the field automorphism of G defined in Proposition 4.2.31. Here, and in the

proofs of results in the next section, we use information about weight multisets for

irreducible K[G]-modules, calculated by Lubeck [58]. Although Lubeck only gives a

single representative of each orbit of the Weyl group contained in a weight multiset,

together with its multiplicity, we can calculate the full orbit using the Magma [11]

computer algebra system, via the code given in Appendix A.3. Each weight in this

orbit has the same multiplicity as the representative, by Lemma 4.2.24. Note that

our labelling of fundamental dominant weights is consistent with that of Magma.


A2V and their dimensions are listed in Table 4.4.2. If p = 3, then Λ(A2V ) is the

disjoint union of [3λ | λ ∈ Φs], [0, 0, 0], and two copies of Φl. Otherwise, Λ(A2V ) is

the disjoint union of [±(1,−1),±(12),±(41),±(43),±(52),±(53)], Φs, [3λ | λ ∈ Φs],

two copies of [2λ | λ ∈ Φs], four copies of Φ, and seven copies of [0]. If p > 3, then

A2V is multiplicity free.

Proof. In each case, adding the elements of Λ(V ) from Lemma 4.4.3 in the way

specified in Lemma 4.2.28 shows that Λ(A2V ) is as required, and the highest weight

of A2V is (63) = 3(21) = 3λ1. Lemma 4.2.37 implies that L(3λ1) is a composition

factor of A2V . When p = 2, L(3λ1) = L(λ1 + 2λ1) ∼= L(λ1) ⊗ L(λ1)φ by Theorem

4.2.40. It follows from Lemmas 4.2.28, 4.2.32 and 4.4.3 that C := Λ(L(3λ1)) is the

disjoint union of [±(1,−1),±(12),±(41),±(43),±(52),±(53)], [3λ | λ ∈ Φs], Φ and

Φl. Thus the highest weight of Λ(A2V )\C is (42) = 2λ1, with multiplicity 2. Hence

the composition factors of A2V include two copies of L(2λ1) ∼= L(λ1)φ, each with

weight multiset [2λ | λ ∈ Φs]. The remaining highest weight is (32) = λ2, with

multiplicity 2, corresponding to two composition factors of L(λ2), each with weight

multiset Φ ∪ [0, 0] by Lemma 4.4.3. The subset of Λ(A2V ) that we have not yet

accounted for is the disjoint union of two copies of Φs and three copies of [0]. It

follows from Lemma 4.4.2 that the remaining composition factors of A2V are two

copies of L(λ1) and three copies of L(0). The composition factors in the p = 3 case

follow via these same methods, and the dimensions of the composition factors in

each case follow from Corollary 4.2.27.

Now, when p > 3, L(3λ1) has dimension 77 [56, Appendix A.49], and the weight

multiset C for this composition factor is such that Λ(A2V )\C is equal to Φ∪[0, 0] [58].

We therefore have from Lemmas 4.2.37 and 4.4.3 that the remaining composition

factor of A2V is L(λ2), with dimension 14. Since the composition factors of A2V

are the same for all p > 3, in terms of dimensions and highest weights, and since

these are pairwise non-isomorphic, Theorem 4.3.5 implies that A2V is multiplicity

free when p > 3.

4.5. The remaining exceptional groups of Lie type 73

Table 4.4.2: The composition factors of A2V and their dimensions and multiplicities,with G = G2 and V = L(λ2).


p = 2

L(3λ1) ∼= L(λ1)⊗ L(λ1)φ 36 1L(2λ1) ∼= L(λ1)φ 6 2

L(λ2) 14 2L(λ1) 6 2L(0) 1 3

p = 3L(3λ1) ∼= L(λ1)φ 7 1

L(λ2) 7 2

p > 3L(3λ1) 77 1L(λ2) 14 1

4.5 The remaining exceptional groups of Lie type

In this section, we look at the linear algebraic groups corresponding to the re-

maining exceptional groups of Lie type. In particular, we will prove similar results

to those in Section 4.4, but in less detail. Except where stated otherwise, the results

here are our own. Note that in almost all cases where G is a linear algebraic group

corresponding to a classical group of Lie type, the composition factors of A2(L(λ1))

follow readily from a paper by Liebeck [55, §1].

For each case that we explore in this section, the roots of G and the fundamen-

tal dominant weights of the corresponding maximal torus are calculated using the

Magma code in Appendix A.3. The dimensions and weight multisets for irreducible

K[G]-modules with p-restricted dominant weights are given by Lubeck1 [56, 58],

with full Weyl orbits calculated using additional Magma code from Appendix A.3.

We also use Theorem 4.2.40, Lemmas 4.2.28 and 4.2.32 and Corollary 4.2.27, to-

gether with Magma, to calculate the weight multisets for irreducible K[G]-modules

whose dominant weights are not p-restricted, and their dimensions. However, we

only describe the weight multiset for a K[G]-module when it is easy to describe in

terms of the root system of G. Finally, we use Theorem 4.3.5 to determine that

some of these K[G]-modules are multiplicity free.

Recall that, in each case, if G = Y`, then the rank of G is `. Also note that

when G = B2, the definitions of the fundamental dominant weights λ1 and λ2

are switched between our notation and Lubeck’s notation, as are the fundamental

dominant weights λ2 and λ3 when G = D4. However, our notation is consistent

with Lubeck’s for the remaining linear algebraic groups corresponding to exceptional

groups of Lie type.

We first consider the case G = B2.

1Lubeck gives this data for all such K[G]-modules whose dimensions are bounded by a partic-ular value depending on G. In fact, for most groups G, the online data [58] has a higher boundthan that of the paper [56], and some classical groups are only considered in the online data.


α1 + 2α2α1

−α1 − 2α2 −α1

α1 + α2

−α2

−α1 − α2

α2

Figure 4.5.1: The roots of the linear algebraic group G = B2.

Proposition 4.5.1. There exists a base ∆ = α1, α2 such that the root system

Φ of G is the union of the short roots Φs := ±(01),±(11) and the long roots

Φl := ±(10),±(12). Here, the fundamental dominant weights of the corresponding

maximal torus are λ1 = (11) and λ2 = (12, 1).

Here, Φs and Φl were determined using Magma. In fact, the short roots of Φ

can be considered as the midpoints of edges of a square in R2 centred at the origin,

with the long roots corresponding to the vertices of the square [73, p. 33–34]. This

is illustrated in Figure 4.5.1. In addition, it is clear that λ2 < λ1. Note that we can

also consider G to be the group C2 [41, p. 23], but in this case, the labelling of the

fundamental dominant weights of G is reversed [56, Appendix A.1].

Lemma 4.5.2. For i ∈ 1, 2, Table 4.5.1 gives the dimension of the irreducible

K[G]-module L(λi) for each value of p. Furthermore, the weight multiset for L(λ1)

is Φs when p = 2, and Φs ∪ [0] otherwise.

Table 4.5.1: The dimensions of irreducible K[G]-modules, with G = B2.

K[G]-module Condition on p Dimension

L(λ1)p = 2 4p > 2 5

L(λ2) All p 4

We are mostly interested in the case with p = 2, as this is the only value of p for

which the group 2B2(q) is a simple group of Lie type. Note that when p = 2, the

Lie algebra of G is a reducible module [60, Theorem 15.20].

In the following theorems, the composition factors are determined by calculating

the weight multisets for the relevant exterior squares using Lemma 4.2.28, and by

recursively applying Lemma 4.2.37 with the aid of Lubeck’s data and Magma calcu-

lations. This also applies to the corresponding results for the other linear algebraic

groups later in this section.

Theorem 4.5.3. Let V ∈ L(λ1), L(λ2). Then for each value of p, the composition

factors of A2V and their dimensions are listed in Tables 4.5.2 and 4.5.3. When


p > 2, A2(L(λ1)) is irreducible, and A2(L(λ2)) is multiplicity free. For V = L(λ1),

the weight multiset for A2V is the union of Φl and [0, 0] when p = 2, and the union

of Φ and [0, 0] when p > 2.

Table 4.5.2: The composition factors of A2V and their dimensions and multiplicities,with G = B2 and V = L(λ1).


p = 2L(2λ2) ∼= L(λ2)φ 4 1

L(0) 1 2p > 2 L(2λ2) 10 1

Table 4.5.3: The composition factors of A2V and their dimensions and multiplicities,with G = B2 and V = L(λ2).


p = 2L(λ1) 4 1L(0) 1 2

p > 2L(λ1) 5 1L(0) 1 1

Note that the irreducibility of A2(L(λ1)) when p > 2 follows from this module

having a single composition factor. Furthermore, Lemma 4.2.25 implies that in

this case, A2(L(λ1)) = L(2λ2) is the Lie algebra of G. In fact, Table 4.5.3 can be

derived from [55, §1], with the fundamental dominant weight λ2 associated with B2

considered as the fundamental dominant weight λ1 associated with C2.

We now consider G = F4. From now on, we will not give geometric descriptions

for the roots of the linear algebraic group G.

Proposition 4.5.4. There exists a base ∆ = α1, α2, α3, α4 such that the root

system Φ of G is the union of the short roots

Φs := ± (0001),±(0010),±(0011),±(0110),±(0111),±(0121),

± (1110),±(1111),±(1121),±(1221),±(1231),±(1232)

and the long roots

Φl := ± (0100),±(0120),±(0122),±(1000),±(1100),±(1120),

± (1122),±(1220),±(1222),±(1242),±(1342),±(2342).

Here, the fundamental dominant weights of the corresponding maximal torus are

λ1 = (2342), λ2 = (3684), λ3 = (2463), and λ4 = (1232).

Observe that λ4 < λ1 < λ3 < λ2.


Lemma 4.5.5. For i ∈ 1, 2, 3, 4, Table 4.5.4 gives the dimension of the irreducible


is Φl ∪ [0, 0] when p = 2, and Φ∪ [0, 0, 0, 0] otherwise. The weight multiset for L(λ4)

is Φs ∪ [0] when p = 3, and Φs ∪ [0, 0] otherwise.

Table 4.5.4: The dimensions of irreducible K[G]-modules, with G = F4.


L(λ1)p = 2 26p > 2 52

L(λ2)p = 2 246p = 3 1222p > 3 1274

L(λ3)p = 2 246p = 3 196p > 3 273

L(λ4)p = 3 25p 6= 3 26

When p > 2, Lemma 4.2.25 implies that L(λ1) is the Lie algebra of G.

Theorem 4.5.6. Let V ∈ L(λ1), L(λ3), L(λ4). Then for each value of p, the

composition factors of A2V and their dimensions are listed in Tables 4.5.5–4.5.7. If

V ∈ L(λ1), L(λ4) and p > 3, then A2V is multiplicity free.

The same methods can be used to determine the composition factors of A2V

when V = L(λ2) and p = 2. However, at the time of writing, Lubeck’s data does

not extend to irreducible K[G]-modules of high enough dimension to facilitate the

calculation of the composition factors of A2V when V = L(λ2) and p > 2.

Table 4.5.5: The composition factors of A2V and their dimensions and multiplicities,with G = F4 and V = L(λ1).


p = 2

L(λ2) 246 1L(2λ4) ∼= L(λ4)φ 26 1

L(λ1) 26 2L(0) 1 1

p = 3L(λ2) 1222 1L(λ1) 52 2

p > 3L(λ2) 1274 1L(λ1) 52 1




p = 2

L(λ2 + λ4) 6396 1L(λ1 + 2λ4) ∼= L(λ1)⊗ L(λ4)φ 676 2

L(λ1 + λ3) 6396 2L(2λ1) ∼= L(λ1)φ 26 3

L(3λ4) ∼= L(λ4)⊗ L(λ4)φ 676 2L(λ2) 246 6

L(λ1 + λ4) 676 8L(2λ4) ∼= L(λ4)φ 26 8

L(λ3) 246 2L(λ1) 26 16L(λ4) 26 6L(0) 1 9

p = 3

L(λ2 + λ4) 11907 1L(λ1 + 2λ4) 6707 1

L(λ3) 196 2L(λ1) 52 2

p = 7

L(λ2 + λ4) 19005 1L(λ1 + 2λ4) 10829 1L(λ3 + λ4) 2991 1L(λ2) 1274 1

L(λ1 + λ4) 1053 2L(λ3) 273 3L(λ1) 52 2

p = 13

L(λ2 + λ4) 15505 1L(λ1 + 2λ4) 10829 1L(λ3 + λ4) 3773 2L(λ2) 1274 1

L(λ1 + λ4) 1053 1L(2λ4) 323 1L(λ3) 273 2L(λ1) 52 1

p /∈ 2, 3, 7, 13

L(λ2 + λ4) 19278 1L(λ1 + 2λ4) 10829 1L(λ3 + λ4) 4096 1L(λ2) 1274 1

L(λ1 + λ4) 1053 1L(λ3) 273 2L(λ1) 52 1

The next group that we consider is G = D4.

Proposition 4.5.7. There exists a base ∆ = α1, α2, α3, α4 such that the root

system Φ of G is

± (0001),±(0010),±(0100),±(0101),±(0110),±(0111),

± (1000),±(1100),±(1101),±(1110),±(1111),±(1211).




p = 2

L(λ3) 246 1L(λ1) 26 1L(λ4) 26 2L(0) 1 1

p = 3L(λ3) 196 1L(λ1) 52 2

p > 3L(λ3) 273 1L(λ1) 52 1


λ1 = (1, 1, 12, 1

2), λ2 = (1211), λ3 = (1

2, 1, 1, 1

2), and λ4 = (1

2, 1, 1

2, 1).

Magma calculations show that Φ is an orbit of the Weyl group, and hence Propo-

sition 4.2.15 implies that all roots in Φ have the same length. Observe that for all

distinct i, j ∈ 1, 3, 4, we have λi < λ2 and λi 6≤ λj.

Lemma 4.5.8. For i ∈ 1, 2, 3, 4, Table 4.5.8 gives the dimension of the irreducible


is Φ ∪ [0, 0] when p = 2, and Φ ∪ [0, 0, 0, 0] otherwise.

Table 4.5.8: The dimensions of irreducible K[G]-modules, with G = D4, and withi ∈ 1, 3, 4.


L(λi) All p 8

L(λ2)p = 2 26p > 2 28

By Lemma 4.2.25, L(λ2) is the Lie algebra of G when p > 2.

Theorem 4.5.9. Let V ∈ L(λ1), L(λ2), L(λ3), L(λ4). Then for each value of p,

the composition factors of A2V and their dimensions are listed in Tables 4.5.9 and

4.5.10. The module A2V is irreducible when V ∈ L(λ1), L(λ3), L(λ4) and p > 2,

and multiplicity free when V = L(λ2) and p > 3.

Table 4.5.9: The composition factors of A2V and their dimensions and multiplicities,with G = D4 and V ∈ L(λ1), L(λ3), L(λ4).


p = 2L(λ2) 26 1L(0) 1 2

p > 2 L(λ2) 28 1


Table 4.5.10: The composition factors of A2V and their dimensions and multiplici-ties, with G = D4 and V = L(λ2).


p = 2

L(λ1 + λ3 + λ4) 246 1L(2λ1) ∼= L(λ1)φ 8 1L(2λ3) ∼= L(λ3)φ 8 1L(2λ4) ∼= L(λ4)φ 8 1

L(λ2) 26 2L(0) 1 3

p = 3L(λ1 + λ3 + λ4) 322 1

L(λ2) 28 2

p > 3L(λ1 + λ3 + λ4) 350 1

L(λ2) 28 1

Note that Table 4.5.9 can be derived from [55, §1].

Later in this chapter, we will show that the irreducibility of the exterior square

of each of L(λ1), L(λ3) and L(λ4) when p > 2 implies that a corresponding G-

module defined over a finite field is irreducible. While we will not be able to use

Theorem 3.2.10 to construct a p-group related to the G-modules corresponding to

L(λ1), L(λ3) and L(λ4), finding an associated reducible Lie power is still of interest.

Since L3V ∼= (A2V ⊗ V )/A3V when p > 3 by Lemma 2.11.8, we will explore the

structure of (A2V ⊗ V )/A3V , with V ∈ L(λ1), L(λ3), L(λ4). Lemmas 4.2.28 and

4.2.29 allow us to calculate the weight multiset of each module here, and we can

then use the same method as above to determine the module’s composition factors.


composition factors of (A2V ⊗ V )/A3V and their dimensions are listed in Table

4.5.11. In each case, (A2V ⊗ V )/A3V is multiplicity free when p /∈ 3, 7.

Table 4.5.11: The composition factors of X := (A2V ⊗V )/A3V and their dimensionsand multiplicities, with G = D4, i, j, k = 1, 3, 4, and V = L(λi).


p = 3L(λi + λ2) 104 1L(λj + λk) 56 1L(λi) 8 1

p = 7L(λi + λ2) 152 1L(λi) 8 2

p /∈ 3, 7 L(λi + λ2) 160 1L(λi) 8 1

Next, we consider G = E6.


λ4

λ3 λ5

λ1 λ2 λ6

Figure 4.5.2: The partial order lattice for the fundamental dominant weights asso-ciated with E6, where an edge connecting an upper weight λi to a lower weight λjmeans that λj < λi.

Proposition 4.5.11. There exists a base ∆ = α1, . . . , α6 such that the root

system Φ of G is

± (000001),±(000010),±(000011),±(000100),±(000110),±(000111),

± (001000),±(001100),±(001110),±(001111),±(010000),±(010100),

± (010110),±(010111),±(011100),±(011110),±(011111),±(011210),

± (011211),±(011221),±(100000),±(101000),±(101100),±(101110),

± (101111),±(111100),±(111110),±(111111),±(111210),±(111211),

± (111221),±(112210),±(112211),±(112221),±(112321),±(122321).


λ1 = (43, 1, 5

3, 2, 4

3, 2

3), λ2 = (122321), λ3 = (5

3, 2, 10

3, 4, 8

3, 4

3), λ4 = (234642), λ5 =

(43, 2, 8

3, 4, 10

3, 5

3), and λ6 = (2

3, 1, 4

3, 2, 5

3, 4

3).

As in the previous case, Φ is an orbit of the Weyl group, and hence Proposition

4.2.15 implies that all roots in Φ have the same length. Note that there are fun-

damental dominant weights λi and λj such that λi 66 λj and λj 66 λi. Figure 4.5.2

illustrates the partial ordering of the fundamental dominant weights. The partial

order relations λ2 < λ1 + λ6 < λ4 < λ1 + λ3 and λ4 < λ5 + λ6 are also relevant to

results that we will state shortly.

Lemma 4.5.12. For i ∈ 1, 2, 3, 4, 5, 6, Table 4.5.12 gives the dimension of the

irreducible K[G]-module L(λi) for each value of p. We have L(λ6) ∼= (L(λ1))∗ and

L(λ5) ∼= (L(λ3))∗. Furthermore, the weight multiset for L(λ2) is the disjoint union

of Φ and 5 copies of [0] when p = 3, and the disjoint union of Φ and 6 copies of [0]

otherwise.

The isomorphisms between modules given in this lemma follow from [56, Ap-

pendix A.3]. When p 6= 3, L(λ2) is the Lie algebra of G by Lemma 4.2.25.


composition factors of A2V and their dimensions are listed in Tables 4.5.13 and

4.5.14. The module A2V is irreducible when V ∈ L(λ1), L(λ6) and p > 2, and

multiplicity free when V = L(λ2) and p > 3.


Table 4.5.12: The dimensions of irreducible K[G]-modules, with G = E6.

K[G]-module(s) Condition on p Dimension

L(λ1), L(λ6) All p 27

L(λ2)p = 3 77p 6= 3 78

L(λ3), L(λ5)p = 2 324p > 2 351

L(λ4)p = 2 1702p = 3 2771p > 3 2925

Table 4.5.13: The composition factors of A2U and A2V and their dimensions andmultiplicities, with G = E6, U = L(λ1), and V = L(λ6).

Condition on pCompositionfactor of A2U

Compositionfactor of A2V

Dimension Multiplicity

p = 2L(λ3) L(λ5) 324 1L(λ6) L(λ1) 27 1

p > 2 L(λ3) L(λ5) 351 1

The dimensions of the K[G]-modules L(λ3), L(λ4) and L(λ5) are too high for

us to calculate the composition factors of the exterior squares. As in the previous

case of G = D4, we would like to find reducible Lie powers of the vector spaces

L(λ1) and L(λ6), whose second Lie powers are irreducible. This will allow us to

later determine the structure of the corresponding G-modules, and then to apply

Theorem 3.2.10 in order to construct a p-group P such that G is a subgroup of

relatively small index in the group A(P ) induced by Aut(P ) on P/Φ(P ). Hence we

determine the composition factors of (A2V ⊗V )/A3V , with V ∈ L(λ1), L(λ6). As

above, this module can be identified with L3V when p > 3.

Table 4.5.14: The composition factors of A2V and their dimensions and multiplici-ties, with G = E6 and V = L(λ2).


p = 2

L(λ4) 1702 1L(λ1 + λ6) 572 2L(λ2) 78 2L(0) 1 1

p = 3L(λ4) 2771 1L(λ2) 77 2L(0) 1 1

p > 3L(λ4) 2925 1L(λ2) 78 1

Theorem 4.5.14. Let V ∈ L(λ1), L(λ6). Then for each value of p, the composi-

tion factors of (A2V ⊗ V )/A3V and their dimensions are listed in Table 4.5.15. In

each case, (A2V ⊗ V )/A3V is multiplicity free when p > 5.


Table 4.5.15: The composition factors of X := (A2U ⊗ U)/A3U and Y :=(A2V ⊗ V )/A3V and their dimensions and multiplicities, with G = E6, U = L(λ1),and V = L(λ6).

Condition on pCompositionfactor of X

Compositionfactor of Y

Dimension Multiplicity

p = 2L(λ1 + λ3) L(λ5 + λ6) 5824 1L(λ1 + λ6) L(λ1 + λ6) 572 1L(λ2) L(λ2) 78 2

p = 3

L(λ1 + λ3) L(λ5 + λ6) 2404 1L(λ4) L(λ4) 2771 1

L(λ1 + λ6) L(λ1 + λ6) 572 2L(λ2) L(λ2) 77 3L(0) L(0) 1 2

p = 5L(λ1 + λ3) L(λ5 + λ6) 5746 1L(λ1 + λ6) L(λ1 + λ6) 650 1L(λ2) L(λ2) 78 2

p > 5L(λ1 + λ3) L(λ5 + λ6) 5824 1L(λ1 + λ6) L(λ1 + λ6) 650 1L(λ2) L(λ2) 78 1

Note that in each row of Tables 4.5.13 and 4.5.15, the composition factor in

the third column is the dual of the composition factor in the second column [56,

Appendix A.3].

We now consider the case G = E7. Here, and in the subsequent case of E8, there

are too many roots for us to list, but these roots can again be calculated using the

Magma code in Appendix A.3.

Proposition 4.5.15. There exists a maximal torus ofG and a base ∆ = α1, . . . , α7such that the fundamental dominant weights of the maximal torus are:

λ1 = (2234321), λ2 = (2,7

2, 4, 6,

9

2, 3,

3

2), λ3 = (3468642), λ4 = (4, 6, 8, 12, 9, 6, 3),

λ5 = (3,9

2, 6, 9,

15

2, 5,

5

2), λ6 = (2346542), and λ7 = (1,

3

2, 2, 3,

5

2, 2,

3

2).

Here, the root system Φ of G contains 126 roots.

Again, Φ is an orbit of the Weyl group, which means that all roots in Φ have the

same length, and the fundamental dominant weights cannot all be ordered according

to the partial order 6. The partial ordering of these weights is shown in Figure 4.5.3.

Note also that λ2 < λ1 + λ7 < λ5 < λ6 + λ7.

Lemma 4.5.16. For i ∈ 1, 2, 3, 5, 6, 7, Table 4.5.16 gives the dimension of the

irreducible K[G]-module L(λi) for each value of p. Furthermore, the weight multiset

for L(λ1) is the disjoint union of Φ and 6 copies of [0] when p = 2, and the disjoint

union of Φ and 7 copies of [0] otherwise.


λ4

λ5

λ3

λ6 λ2

λ1 λ7

Figure 4.5.3: The partial order lattice for the fundamental dominant weights asso-ciated with E7, where an edge connecting an upper weight λi to a lower weight λjmeans that λj < λi.



L(λ1)p = 2 132p > 2 133

L(λ2)p = 3 856p 6= 3 912

L(λ3)p = 2 7106p = 3 8512p > 3 8645

L(λ5)p = 2 21184p = 3 25896p > 3 27664

L(λ6)p = 2 1274p = 7 1538

p /∈ 2, 7 1539L(λ7) All p 56

We have from Lemma 4.2.25 that L(λ1) is the Lie algebra of G when p > 2.

Note that Lubeck [56, 58] does not give the dimension of L(λ4), which means that

it is higher than 100000 for all p. Indeed, Lemma 4.3.1 implies that for almost all

values of p, the dimension of L(λ4) is equal to the dimension of the corresponding

module defined over C, which is 365750 [27, p. 531].

Theorem 4.5.17. Let V ∈ L(λ1), L(λ7). Then for each value of p, the composi-

tion factors of A2V and their dimensions are listed in Tables 4.5.17 and 4.5.18. The

module A2(L(λ1)) is multiplicity free when p > 3, while A2(L(λ7)) is multiplicity

free when p /∈ 2, 7.

The dimensions of the other K[G]-modules listed in Lemma 4.5.16 are too high

for us to calculate the composition factors of the exterior squares. However, for

each p, if V = L(λ2), then the highest weight of A2V is λ4. This means that the

dimension of L(λ4) is at most dim(A2V ), which is 356940 when p = 3 and 415416

otherwise, by Lemma 2.7.3.




p = 2

L(λ3) 7106 1L(λ6) 1274 1L(λ1) 132 2L(0) 1 2

p = 3L(λ3) 8512 1L(λ1) 133 2

p > 3L(λ3) 8645 1L(λ1) 133 1



p = 2L(λ6) 1274 1L(λ1) 132 2L(0) 1 2

p = 7L(λ6) 1538 1L(0) 1 2

p /∈ 2, 7 L(λ6) 1539 1L(0) 1 1

Even though the G-module A2V with V = L(λ7) is reducible for all p, we will

see later in this thesis that in order to use Theorem 3.2.10 to construct a p-group P

such that G is a subgroup of A(P ) of relatively small index, we must determine the

submodule structure of L3V . We can again identify this module with (A2V⊗V )/A3V

when p > 3.

Theorem 4.5.18. Let V = L(λ7). Then for each value of p, the composition factors

of (A2V ⊗ V )/A3V and their dimensions are listed in Table 4.5.19. The module

(A2V ⊗ V )/A3V is multiplicity free when p /∈ 2, 3, 7, 11, 19.

Finally, we consider G = E8.

Proposition 4.5.19. There exists a maximal torus ofG and a base ∆ = α1, . . . , α8such that the fundamental dominant weights of the maximal torus are:

λ1 = (4, 5, 7, 10, 8, 6, 4, 2), λ2 = (5, 8, 10, 15, 12, 9, 6, 3), λ3 = (7, 10, 14, 20, 16, 12, 8, 4),

λ4 = (10, 15, 20, 30, 24, 18, 12, 6), λ5 = (8, 12, 16, 24, 20, 15, 10, 5),

λ6 = (6, 9, 12, 18, 15, 12, 8, 4), λ7 = (4, 6, 8, 12, 10, 8, 6, 3), and λ8 = (23465432).

Here, the root system Φ of G contains 240 roots.

Once again, Φ is an orbit of the Weyl group, and hence all roots in Φ have the

same length. Here, λ8 < λ1 < λ7 < λ2 < λ6 < λ3 < λ5 < λ4.


Table 4.5.19: The composition factors of (A2V ⊗ V )/A3V and their dimensions andmultiplicities, with G = E7 and V = L(λ7).


p = 2

L(λ6 + λ7) 50160 1L(λ1 + λ7) 6480 1L(λ2) 912 2L(λ7) 56 1

p = 3

L(λ6 + λ7) 24264 1L(λ5) 25896 1

L(λ1 + λ7) 6480 1L(λ2) 856 2L(λ7) 56 3

p = 7

L(λ6 + λ7) 51072 1L(λ1 + λ7) 5568 1L(λ2) 912 2L(λ7) 56 1

p = 11

L(λ6 + λ7) 44592 1L(λ1 + λ7) 6480 2L(λ2) 912 1L(λ7) 56 1

p = 19

L(λ6 + λ7) 51072 1L(λ1 + λ7) 6424 1L(λ2) 912 1L(λ7) 56 2

p /∈ 2, 3, 7, 11, 19

L(λ6 + λ7) 51072 1L(λ1 + λ7) 6480 1L(λ2) 912 1L(λ7) 56 1

Lemma 4.5.20. For i ∈ 1, 2, 7, 8, Table 4.5.20 gives the dimension of the irre-

ducible K[G]-module L(λi) for each value of p. Furthermore, the weight multiset

for L(λ8) is the disjoint union of Φ and 8 copies of [0], for all p.



L(λ1)p = 2 3626p > 2 3875

L(λ2)

p = 2 143376p = 3 113243p = 7 147002

p /∈ 2, 3, 7 147250

L(λ7)p = 2 26504

p ∈ 3, 5 30132p > 5 30380

L(λ8) All p 248

Lemma 4.2.25 implies that, for all p, L(λ8) is the Lie algebra of G. Note that


Lubeck gives the dimension of L(λ2) in his online data [58], but not in his original

paper [56]. Furthermore, Lubeck does not give the dimension of L(λi) for any

i ∈ 3, 4, 5, 6, and hence the dimension of each of these K[G]-modules is higher

than 500000. By Lemma 4.3.1, the dimension of L(λi) for a given i is equal to

that of the corresponding module defined over C for almost all values of p. This

dimension is 6696000, 6899079264, 146325270 or 2450240, for i equal to 3, 4, 5 or 6,

respectively [27, p. 531].


A2V and their dimensions are listed in Table 4.5.21. When p > 5, A2V is multiplicity

free.



p = 2

L(λ7) 26504 1L(λ1) 3626 1L(λ8) 248 2L(0) 1 2

p ∈ 3, 5 L(λ7) 30132 1L(λ8) 248 2

p > 5L(λ7) 30380 1L(λ8) 248 1

The dimensions of the other K[G]-modules listed in Lemma 4.5.20 are too high

for us to calculate the composition factors of the exterior squares. However, for each

p and each i ∈ 1, 2, if V = L(λi), then the highest weight ofA2V is λi+2. Therefore,

in each case, dim(A2V ) is an upper bound for dim(L(λi+2)). Note that the highest

weight of A2(L(λ7)) is λ6 + λ8, and Lubeck also does not give dim(L(λ6 + λ8)).

We conclude this section by describing the minimal modules over K of the linear

algebraic groups corresponding to exceptional groups of Lie type, including the

group G2. Table 4.5.22 gives the highest weight and dimension of each minimal

module, up to isomorphism and twisting by a field automorphism of G, based on

Lubeck’s [56] lists of irreducible K[G]-modules. This table is separated into two

sections, with the second consisting of the linear algebraic groups that correspond to

exceptional twisted groups of Lie type, but not to any exceptional Chevalley group.

Note that the highest weight of each module in this table is a fundamental dominant

weight, and that we have determined the composition factors of the exterior square

of each of these modules. It follows from the definition of a module twisted by an

automorphism that if α ∈ Aut(G) and if V is a K[G]-module, then the composition

factors of V α are obtained by twisting the composition factors of V by α.

Observe that if G 6= D4, and if U and V are two distinct K[G]-modules in Ta-

ble 4.5.22 for a fixed prime p, then G has a graph automorphism of order two by

4.6. Finite groups 87

Table 4.5.22: The highest weights and dimensions of minimal K[G]-modules, upto isomorphism and twisting by a field automorphism of G, with G an exceptionalgroup of Lie type.

G Highest weight Condition on p Dimension

G2 λ1p = 2 6

p > 2 7

G2 λ2 p = 3 7

F4 λ1 p = 2 26

F4 λ4p = 3 25

p 6= 3 26

E6 λ1 All p 27

E6 λ6 All p 27

E7 λ7 All p 56

E8 λ8 All p 248

B2 λ1 p = 2 4

B2 λ2 All p 4

D4 λ1 All p 8

D4 λ3 All p 8

D4 λ4 All p 8

Proposition 4.2.36, and there is an associated permutation of fundamental domi-

nant weights. In fact, this permutation interchanges the highest weight of U and

the highest weight of V [53, p. 180, p. 191–192]. Similarly, if G = D4, then the

permutation associated with the graph automorphism of G of order 3 transitively

permutes the highest weights of the three distinct K[G]-modules in Table 4.5.22.

Thus Theorem 4.2.33 and Proposition 4.2.36 yield the following.

Proposition 4.5.22. Let G be an exceptional group of Lie type, and fix the prime

p. In addition, let γ be a graph automorphism of G, with |γ| = 3 if G = D4. Suppose

also that U = L(λ) and V = L(µ) are minimal K[G]-modules, where λ and µ are

distinct fundamental dominant weights. Then Uγ ∼= V or V γ ∼= U (or both).

4.6 Finite groups

In this section, we explore modules for the finite groups G = tY`(q) and G.

Recall that if (t, Y ) = (2, B), then we call G a Suzuki group, and that if (t, Y ) ∈(2, G), (2, F ), then we call G a Ree group. All results in this section are our own,

unless stated otherwise.

By Definition 4.2.19, each fundamental dominant weight λi of T is defined with

respect to a corresponding root αi ∈ ∆. Furthermore, Propositions 4.4.1, 4.5.1 and

4.5.4 imply that the root system of each G ∈ B2, G2, F4 contains roots of two

different lengths. For each of these linear algebraic groups, we can therefore define


the set L of irreducible K[G]-modules L(λ) such that λ is a linear combination∑ì=1 ciλi of fundamental dominant weights of T , with 0 6 ci 6

√q/p− 1 whenever

αi is a long root, and with 0 6 ci 6√qp − 1 whenever αi is a short root. If G

is any other linear algebraic group, then we define L to be the set of irreducible

K[G]-modules L(λ) such that λ is an element of X(q), defined in Definition 4.2.39.

Lubeck [56, Theorem 2.3] presents the following theorem involving the set L.

Note that Lubeck’s definition of L in the first case above appears different to our

definition, as Lubeck’s parameter “q” in this case is the square of our parameter q.

Theorem 4.6.1 (Steinberg). The irreducible K[G]-modules are the restrictions to

G of the irreducible K[G]-modules in L. Furthermore, if L(λ), L(µ) ∈ L, then the

restrictions of L(λ) and L(µ) to G are isomorphic if and only if λ = µ.

The above theorem and Examples 4.2.12 and 4.2.20 imply that when G =

PSL(2, q) with q > 3, there is a 1-1 correspondence between the set of distinct ir-

reducible K[SL(2, q)]-modules and the set of irreducible K[SL(2, K)]-modules L(i),

for i ∈ 0, 1, . . . , q − 1. Thus there are q distinct irreducible K[SL(2, q)]-modules.

Corollary 4.6.2. Let V be a K[G]-module whose composition factors all lie in L.

Then the composition factors of V |G are the restrictions to G of the composition

factors of V .

Proof. Let 0 = V0 ⊂ V1 ⊂ · · · ⊂ Vn = V be a composition series for V . Then

0 = V0|G ⊂ V1|G ⊂ · · · ⊂ Vn|G = V |G is a chain of submodules of V |G. We

also have (Vi/Vi−1)|G ∼= Vi|G/Vi−1|G by the definition of the action of G on each of

these quotient modules. Hence Vi|G/Vi−1|G is the restriction to G of the composition

factor Vi/Vi−1 ∈ L. Furthermore, this restriction is irreducible by Theorem 4.6.1,

and thus it is a composition factor of V |G. In particular, the restricted chain of

submodules is a composition series for V |G, and the result follows.

For the following results, recall the notation G = tY`(q).

Proposition 4.6.3 (Steinberg [53, Proposition 5.4.4]). If G is a Suzuki or Ree group,

then Fq is a splitting field for G. Otherwise, Fqt is a splitting field for G.

Theorem 4.6.4. Let u := 1 if G is a Suzuki or Ree group, and otherwise let

u := t. Up to isomorphism, there is a 1-1 correspondence between irreducible K[G]-

modules in L and irreducible Fqt [G]-modules. Specifically, if U ∈ L, and if V is the

corresponding irreducible Fqt [G]-module, then U |G is isomorphic to theK[G]-module

V K constructed by extending the scalars. More generally, let V be an Fqu [G]-

module that is constructed from irreducible Fqu [G]-modules via tensor products,

exterior powers and quotients (of submodules constructed similarly). Then the

composition factors of V correspond to the composition factors of the K[G]-module

W constructed via the same operations on the corresponding elements of L, and

W |G ∼= V K . If W is multiplicity free, then so are W |G and V .


Proof. Since Fqu is a splitting field for G by Proposition 4.6.3, Proposition 2.9.16

tells us that there is a 1-1 correspondence Vi ←→ V Ki between distinct irreducible

Fqu [G]-modules and distinct irreducible K[G]-modules. Theorem 4.6.1 therefore

implies that the restrictions to G of distinct elements of L are obtained from the

distinct irreducible Fqu [G]-modules by extending the scalars.

Next, if V is an Fqu [G]-module constructed via tensor products, exterior powers

and quotients of irreducible modules from a set U , then Lemma 2.9.10 shows that the

K[G]-module V K is isomorphic to the K[G]-module constructed by performing the

corresponding operations on the irreducible modules UK with U ∈ U . From above,

each module UK is isomorphic to X|G for some irreducible K[G]-module X ∈ L.

In fact, the module constructed via the aforementioned operations on the modules

X|G is isomorphic to the restriction to G of the K[G]-module W constructed via

these operations on the modules X. This is clear from the definitions of the group

actions associated with these modules. Therefore, W |G ∼= V K .

Now, the composition factors of W |G are the restrictions to G of those of W

by Corollary 4.6.2, and the composition factors of V K are obtained by extending

the scalars of V by Lemma 2.9.18. Hence the above correspondence of irreducible

modules holds between the composition factors of W and the composition factors

of V . If W is multiplicity free, then this correspondence of composition factors

(which preserves non-isomorphism) and Proposition 2.9.1 imply that W |G is also

multiplicity free. In this case, V is multiplicity free by Proposition 2.9.17, as K is

algebraic over Fqu , which is a splitting field for G.

Note that corresponding K[G]-modules and Fqu [G]-modules have equal dimen-

sion.

Example 4.6.5. In the case of G = PSL(2, q), with q > 3, there is a 1-1 corre-

spondence between the set of distinct irreducible Fq[SL(2, q)]-modules and the set

of irreducible K[SL(2, K)]-modules L = L(i) | i = 0, 1, . . . , q − 1. Let V0, V1

and V2 be the irreducible Fq[SL(2, q)]-modules associated with L(0), L(1) and L(2),

respectively. Note that V0 is the trivial irreducible Fq[SL(2, q)]-module. It follows

from Example 4.2.38 that the composition factors of V1 ⊗ V1 are V2 and two copies

of V0, and that the composition factors of A2V1 are two copies of each of V0 and V2.

Now, by Table 4.1.3, if q is even, then the Schur multiplier of G has order 2 if

q = 4 and order 1 otherwise, and if q is odd, then the Schur multiplier of G has

order 6 if q = 9 and order 2 otherwise. Thus this Schur multiplier is a p-group if and

only if q is even. Corollary 4.2.2 therefore implies that SL(2, q) ∼= PSL(2, q) when q

is even, but not when q is odd.

Proposition 4.6.6 ([53, p. 193–194]). Suppose that q = pe for some positive integer

e, and that G is not a Suzuki or Ree group. If t 6= 1, then let γ be the graph

automorphism of G of order t. Additionally, let f be a positive integer, let U be an


absolutely irreducible Fpf [G]-module that cannot be written over a proper subfield

of Fpf , and let W ∈ L be the K[G]-module such that W |G = UK . Then either:

(i) f | e, and there exists an irreducible K[G]-module V such that dim(U) =

dim(V )e/f ; or

(ii) t > 1; W 6∼= W γ; f | te and f - e; and there exists an irreducible K[G]-module

V such that dim(U) = dim(V )te/f .

Proposition 4.6.7 ([53, Remark 5.4.7(b)]). Suppose that q = pe for some positive

integer e, and that G = 2Y`(q) is a Suzuki or Ree group. Additionally, let f be a

positive integer, and let U be an absolutely irreducible Fpf [G]-module that cannot

be written over a proper subfield of Fpf . Then f | e and ne/f 6 dim(U), where

(Y, n) ∈ (B, 4), (G, 7), (F, 26). In particular, if dim(U) < n2, then f = e.

Now, when G ∈ 3D4(q), 2E6(q), we have from Proposition 4.5.22 that the graph

automorphism γ of G interchanges the minimal K[G]-modules from Table 4.5.22.

Hence Proposition 4.6.6 implies that the corresponding absolutely irreducible G-

modules cannot be written over Fq, or over a proper subfield of Fq. Furthermore,

Proposition 4.6.7 implies that when G is a Suzuki or Ree group, the absolutely

irreducible G-modules corresponding to minimal K[G]-modules cannot be written

over a proper subfield of Fq. Recall from Table 4.1.2 that when G is a Suzuki or

Ree group, we have q > p. Thus there is no twisted exceptional group of Lie type

G such that an absolutely irreducible G-module corresponding to a minimal K[G]-

module can be written over Fp. This means that we cannot apply Theorem 3.2.10

or Theorem 3.3.9 to the images of the representations afforded by these modules.

Another reason that we cannot apply these theorems when G ∈ 2B2(q), 2F4(q) is

that q is even in these cases, by Table 4.1.2.

Throughout the remainder of this section, we assume that:

G is an exceptional Chevalley group.

Lemma 4.6.8. Let d be the dimension of a minimal K[G]-module, as in Table

4.5.22. Then:

(i) d is the dimension of the minimal Fq[G]-modules, and there is a unique quasi-

equivalence class Q of these modules;

(ii) each module in Q is absolutely irreducible and faithful, and cannot be written

over any proper subfield of Fq;

(iii) the images of the Fq-representations afforded by modules in Q form a single

conjugacy class of subgroups of GL(d, q);

(iv) if V ∈ Q, and if α is a nontrivial field automorphism of G, then V 6∼= V α;


(v) if G 6= E6(q) and (G, p) /∈ (G2(q), 3), (F4(q), 2), then Q is also the unique

equivalence class of minimal Fq[G]-modules up to isomorphism and twisting

by a field automorphism;

(vi) if G = E6(q) or if (G, p) ∈ (G2(q), 3), (F4(q), 2), then there are two equiva-

lence classes of minimal Fq[G]-modules with respect to isomorphism and twist-

ing by a field automorphism, and these equivalence classes are interchanged

by the graph automorphism of G; and

(vii) if G = E6(q), then a given minimal Fq[G]-module and its dual lie in differ-

ent equivalence classes with respect to isomorphism and twisting by a field

automorphism.

Proof. First, Theorem 4.6.4 implies that the minimal K[G]-modules and the min-

imal Fq[G]-modules have the same dimension. Table 4.5.22 lists all minimal K[G]-

modules, up to isomorphism and twisting by a field automorphism, where the high-

est weight of each of these modules is a fundamental dominant weight. Since Lis the set of irreducible K[G]-modules whose highest weights lie in X(q), it fol-

lows from Theorem 4.2.40 that each minimal K[G]-module in L can be written as

L(λ)φi ∼= L(piλ), with λ a fundamental dominant weight and i ∈ 0, 1, . . . , e − 1,

where q = pe. Moreover, L(piλ) 6∼= L(pjλ) if i 6= j by Theorem 4.2.33. Observe

that if α is an automorphism of G that induces an automorphism α (with slight

abuse of notation) of G, and if an Fq[G]-module V corresponds to a K[G]-module

W in the way specified in Theorem 4.6.4, then the definitions of the actions of G

and G on V α and Wα, respectively, imply that V α corresponds to Wα. Since the

field automorphisms of G induce those of G by Proposition 4.2.31, Theorem 4.6.4

implies that the distinct (up to isomorphism, or up to isomorphism and twisting

by a field automorphism) minimal Fq[G]-modules are those that correspond to the

distinct (in the same way) minimal K[G]-modules in L. In particular, if G = E6(q)

or if (G, p) ∈ (G2(q), 3), (F4(q), 2), then there are two equivalence classes of these

G-modules up to isomorphism and twisting by a field automorphism. Otherwise,

there is a unique such equivalence class, and hence a unique quasi-equivalence class

of these modules by Definition 2.8.10.

Now, we have from Lemma 4.5.12 that when G = E6(q), the highest weight of

each minimal K[G]-module is not a scalar multiple of the highest weight of its dual,

and hence these modules are not equivalent up to isomorphism and twisting by a field

automorphism. Additionally, in this case, or when (G, p) ∈ (G2(q), 3), (F4(q), 2),Proposition 4.5.22 implies that twisting each minimal K[G]-module by the graph

automorphism of G results in a module that is not equivalent up to isomorphism

and twisting by a field automorphism. We also have from Proposition 4.2.36 that

the graph automorphism of G induces the graph automorphism of G. Hence we see

by considering the definitions of the actions associated with the dual of a module


and with a module twisted by a group automorphism that these duality and twisting

properties also hold for the corresponding irreducible Fq[G]-modules. Therefore, the

aforementioned equivalence classes of Fq[G]-modules form a single quasi-equivalence

class. In all cases, the quasi-equivalence of modules and Lemma 2.8.11 imply the

conjugacy of the afforded representations.

Next, Corollary 4.2.3 implies that unless G ∈ E6(q), E7(q), we can identify G

with the simple group G. In this case, each minimal Fq[G]-module is faithful by

Proposition 2.8.4. In fact, the minimal Fq[G]-modules are faithful even when G ∈E6(q), E7(q) [53, p. 202–203]. Moreover, each minimal Fq[G]-module is absolutely

irreducible by Proposition 4.6.3. Finally, as there are no nontrivial irreducible K[G]-

modules of dimension less than d, Proposition 4.6.6 implies that no minimal Fq[G]-

module can be written over a proper subfield of Fq.

In fact, Theorem 4.6.1 implies that the dimension of a minimal K[G]-module is

equal to the dimension of a minimal K[G]-module.

Proposition 4.6.9. If G 6∼= G, then Z(G) is the only nontrivial proper normal

subgroup of G.

Proof. Since G is quasisimple by Theorem 4.2.4, each proper normal subgroup of G

lies in Z(G) [10, p. 350]. If G is not isomorphic to G, i.e., if G is listed in Corollary

4.2.3, then |Z(G)| is prime [60, Corollary 24.13]. Lagrange’s Theorem therefore

implies that Z(G) is the only nontrivial proper normal subgroup of G.

Proposition 4.6.10. Let d be the dimension of a minimal K[G]-module, as in

Table 4.5.22. Additionally, let H be a nontrivial quotient of G, and let V be a

faithful Fq[H]-module of dimension at most d. Then dim(V ) = d, H = G, and V is

irreducible.

Proof. By Proposition 4.6.9, if G is not simple and H 6= G, then H = G/Z(G),

i.e., H ∼= G by Lemma 4.2.1. Hence in all cases, H is non-abelian. The image of a

1-dimensional Fq-representation is a subgroup of the abelian group GL(1,Fq), and so

the faithful H-module V has dimension at least 2. Suppose that V is reducible. Then

each Fq[H]-composition factor of V is an irreducible Fq[H]-module of dimension less

than d. Observe that each irreducible Fq[H]-module is also an irreducible Fq[G]-

module, where the kernel of the afforded representation of G contains Z(G) ifH ∼= G.

Hence each Fq[H]-composition factor of V is the trivial irreducible module by Lemma

4.6.8. Since V is a vector space over Fq, every Fq[H]-composition series for V is also

a normal series for V , with each group in this series stabilised by H. Furthermore,

as V is faithful, H is a group of automorphisms of the p-group V . This means that

H is a p-group [30, p. 178–179]. However, |H| is divisible by |G|, which is not a

power of p by Table 4.1.3. This is a contradiction, and hence V is irreducible. In

particular, dim(V ) = d by Lemma 4.6.8. This lemma also implies that V is a faithful

Fq[G]-module, and therefore H = G.


Recall from Lemma 2.11.8 that if V is a nonzero module over a field of char-

acteristic not equal to 2, then L2V ∼= A2V , and if the characteristic of the field

is also not equal to 3, then L3V ∼= (A2V ⊗ V )/A3V . We will now describe the

submodule structure of L2V , where V is a minimal Fq[G]-module with p > 2, and

the structure of L3V when G ∈ E6(q), E7(q), with p > 3. In most cases, we also

show that the submodule structure of L2V or L3V is equivalent (in terms of con-

tainments and dimensions) to the submodule structure of the same Lie power of the

corresponding minimal K[G]-module and the corresponding minimal K[G]-module.

As the submodule structure of the exterior square of a module is important in many

applications, we use the notation A2V in the following theorem, instead of L2V .

Theorem 4.6.11. Let V be a minimal Fq[G]-module, with p odd, and let W ∈ Lbe the irreducible K[G]-module corresponding to V , as in Theorem 4.6.4.

(i) The submodule structure of A2V is given in Table 4.6.1, and is equivalent to

the submodule structure of each of A2W and (A2W )|G. If p is an “exceptional

prime” for G, i.e., if (G, p) lies in the set

(G2(q), 3), (F4(q), 3), (E7(q), 7), (E8(q), 3), (E8(q), 5),

then A2V is uniserial. Otherwise, A2V is multiplicity free. In particular, if

G = E6(q), then A2V is irreducible.

(ii) If G ∈ G2(q), E8(q) and B ∈ V,W,W |G, then the quotient of A2B by its

largest maximal submodule is isomorphic to B.

(iii) Suppose that G ∈ E6(q), E7(q) and that p > 3, with q = p if G = E6(q)

and p = 5, or if G = E7(q) and p ∈ 7, 11, 19. The submodule structure of

L3V is given in Figures 4.6.1–4.6.6. If p is not a listed “exceptional prime”

for G, then L3V is multiplicity free, and the submodule structure of L3V is

equivalent to that of L3W and that of (L3W )|G.

(iv) For a fixed group G and a fixed prime p > 3, the dimensions of the composition

factors of L3V do not depend on q or on the choice of V .

Proof. We have from Lemma 4.6.8 that for a given combination of G, d and q as

allowed by this lemma, there is a unique quasi-equivalence class Q of minimal Fq[G]-

modules. Proposition 2.8.23 therefore implies that all modules in A2T | T ∈ Q are

quasi-equivalent, as are all modules in L3T | T ∈ Q. It follows from Corollary

2.8.9 that, in each of these sets, all modules have an equivalent submodule structure,

regardless of the choice of V . Furthermore, if (G, q,W ) 6= (G2(q), 3, L(λ2)), then

the tables in Sections 4.4 and 4.5 show that each composition factor of A2W , or of

L3W when G ∈ E6(q), E7(q), lies in L = X(q). If (G, q,W ) = (G2(q), 3, L(λ2)),

then the only composition factor of A2W that does not lie in L is L(3λ1) ∼= L(λ1)φ.


Recall from Proposition 4.2.31 that the field automorphisms of G induce the field

automorphisms of G, and that G has no nontrivial field automorphism as q = p. It

therefore follows from the definition of the action associated with a module twisted

by a group automorphism that the restriction of L(3λ1) to G is isomorphic to the

restriction of L(λ1) to G. Hence Theorem 4.6.4 implies that, in each case, the com-

position factors of A2V or L3V correspond to those of A2W or L3W , respectively.

In particular, for a fixed group G and a fixed prime p, the dimensions of the com-

position factors of A2V or of L3V do not depend on q or on the choice of V , nor do

the composition factors of A2W , L3W or their restrictions to G.

Suppose now that (G, p) is not in the set

(G2(q), 3), (F4(q), 3), (E7(q), 7), (E8(q), 3), (E8(q), 5).

The module A2W is multiplicity free by the results in Sections 4.4 and 4.5, and this

module is irreducible when G = E6(q). Furthermore, when G = E6(q) and p > 5,

or when G = E7(q) and p /∈ 2, 3, 7, 11, 19, L3W is multiplicity free by Theorem

4.5.14 or 4.5.18, respectively. In each case, Theorem 4.6.4 implies that the corre-

sponding Fq[G]-module A2V or L3V is irreducible or multiplicity free as specified,

as is the restriction of A2W or L3W to G. The required submodule structure of

each K[G]-module, each K[G]-module and each Fq[G]-module here follows from the

composition factors of the K[G]-modules given in Sections 4.4 and 4.5. Specifically,

isomorphism of modules gives a 1-1 correspondence between the composition factors

of each module and its irreducible submodules by Proposition 2.8.17, and these irre-

ducible submodules determine all of the module’s submodules via Corollary 2.8.16.

Next, suppose that G = Y`(q), where

(Y`, p) ∈ (G2, 3), (F4, 3), (E7, 7), (E8, 3), (E8, 5).

Additionally, let X := Y`(p), let X be the simply connected version of X, and let U

be a minimal Fp[X]-module. Then G is the linear algebraic group associated with X

by Proposition 4.2.5. We have (A2U)K ∼= (A2W )|X and (A2V )K ∼= (A2W )|G from

Theorem 4.6.4. In fact, X is a subgroup of G [13, Lemma 5.1.6], and it follows that

(A2U)K is the restriction of (A2V )K to X. Computations using the Magma code

given in Appendix A.4 show that the submodule structure of A2U is as required,

and in particular, that A2U is uniserial. Since K is algebraic over Fp, it follows

from Lemma 2.9.19 that the submodule structure of A2U is equivalent to that of

(A2U)K . Since (A2V )K and (A2U)K have equivalent composition factors (in terms

of dimension), and since (A2U)K is uniserial, the submodule structures of these two

modules are equivalent. For the same reason, A2W has an equivalent submodule

structure to that of its restriction to G, (A2V )K . Furthermore, A2V is uniserial by

Lemma 2.9.19, and again its submodule structure is equivalent to that of (A2V )K .

Now, let G ∈ G2(q), E8(q), and let S be the largest maximal submodule of

A2W . It follows from Table 4.5.22 that Tables 4.4.1, 4.4.2 and 4.5.21 list the com-


position factors of W , with W defined up to isomorphism and twisting by a field

automorphism. These tables and Lemma 4.2.32 imply that either (A2W )/S ∼= W ,

or (G, p) = (G2(q), 3) and the two composition factors of S have the same highest

weight. However, S is uniserial in this case, and hence Proposition 4.3.4 implies that

its two composition factors have different highest weights. Hence (A2W )/S ∼= W . It

is therefore clear that (A2W )|G/S|G ∼= W |G. From above, A2W , (A2W )G∼= (A2V )K

and A2V have equivalent submodule structures. It follows that S|G is the largest

maximal submodule of (A2W )|G, and that S|G ∼= NK , where N is the largest maxi-

mal submodule of A2V . Thus the irreducible module (A2V )K/NK is isomorphic to

(A2W )|G/S|G ∼= W |G. Recall that (A2V )K/NK ∼= ((A2V )/N)K by Lemma 2.9.10.

This means that both V and (A2V )/N correspond to W via the 1-1 correspondence

of Theorem 4.6.4, and so (A2V )/N ∼= V .

Additional Magma computations detailed in Appendix A.4 were used to deter-

mine the submodule structure of L3V in the case where G = E6(5). We also used

Magma in the case where G = E7(7) to show that L3V contains a 56-dimensional

submodule; a 51072-dimensional submodule; and a uniserial 7392-dimensional sub-

module with two proper nonzero submodules, of dimension 6480 and 912, respec-

tively. We will write Uk to denote these submodules, where dim(Uk) = k. It fol-

lows from the dimensions of the composition factors of L3V that U56, U912 and

U51072 are each irreducible. In fact, as U912 is the only irreducible submodule that

lies in U7392, and since the intersection between an irreducible submodule and any

other submodule is the full irreducible submodule or the zero submodule, we have

U56 +U7392 +U51072 = U56⊕U7392⊕U51072. By considering dimensions, using Corol-

lary 2.11.9, we see that this direct sum is equal to L3V . No two of these three

direct summands have a common composition factor, and so Theorem 2.8.19 im-

plies that the submodules of L3V are exactly the direct sums of the submodules of

U56, the submodules of U7392 and the submodules of U51072. Containments between

submodules are also determined by applying Theorem 2.8.19 to each submodule

of L3V . Similar computations were performed in the case of G = E7(19), where

L3V is the direct sum of an irreducible 912-dimensional submodule, an irreducible

51072-dimensional submodule and a uniserial 6536-dimensional submodule. Finally,

in the case of G = E7(11), we used Magma to show that L3V contains exactly three

irreducible submodules, of dimension 56, 912 and 6480, respectively, as well as a

57552-dimensional submodule. By considering the dimensions of the composition

factors of L3V , we see that U6480 is the unique irreducible submodule of U57552. A

final Magma calculation showed that U57552/U6480 has a unique irreducible submod-

ule, of dimension 44592. The given submodule structure of L3V therefore follows

from the Correspondence Theorem, from the dimensions of the composition factors

of L3V , and from the fact that L3V = U56 ⊕ U912 ⊕ U57552.


Table 4.6.1: The submodule structure of A2V , where V is a minimal Fq[G]-module,with q a power of a prime p > 2. Each submodule is labelled by its dimension.

G dExceptional

primesStandard structure

of A2V

Structure of A2V forexceptional primes

G2(q) 7 3

21

7 14

0

21

14

7

0

F4(q)25, p = 326, p > 3

3

325

52 273

0

300

248

52

0

E6(q) 27 None351

0

N/A

E7(q) 56 7

1540

1 1539

0

1540

1539

1

0

E8(q) 248 3, 5

30628

248 30380

0

30628

30380

248

0

6552

728 5902 6474

78 650 5824

0

Figure 4.6.1: The submodule structure of L3V , where V is a minimal Fq[G]-module,with G = E6(q) and q a power of a prime p > 5. Each submodule is labelled by itsdimension.


6552

5902 6474

5824 728

78 650

0

Figure 4.6.2: The submodule structure of L3V , where V is a minimal Fq[G]-module,with G = E6(5). Each submodule is labelled by its dimension.

58520

7448 52040 57608 58464

968 6536 511287392 51984 57552

56 912 6480 51072

0

Figure 4.6.3: The submodule structure of L3V , where V is a minimal Fq[G]-module,with G = E7(q) and q a power of a prime p /∈ 2, 3, 7, 11, 19. Each submodule islabelled by its dimension.

58520

7448 57608 58464

6536 7392 52040 57552

968 6480 51128 51984

56 912 51072

0



58520

57608 52040 58464

51128 57552 7448 51984

6536 51072 968 7392

56 6480 912

0


58520

7448 58464 57608

7392 6536 52040 57552

968 6480 51984 51128

912 56 51072

0



Our proof for the uniserial cases in the above theorem is similar to part of the

proof of [13, Proposition 5.5.3]. Note that the reducibility of the exterior square of an

irreducible 7-dimensional Fq[G2(q)]-module2 was previously investigated by Schroder

[68, Ch. 9.3.2], but in less detail than in Theorem 4.6.11. It is also known that there

exists an irreducible 7-dimensional module over R for the related group G2(R), which

we will define in Section 5.2, whose exterior square is the direct sum of an irreducible

7-dimensional submodule and an irreducible 14-dimensional submodule [34, p. 9].

Observe that, except in the case where G = F4(q) and d depends on whether

or not p is an exceptional prime, the dimensions of the submodules of each G-

module described in Theorem 4.6.11 that is associated with an exceptional prime

are exactly the same as the corresponding module when the prime is not exceptional.

Furthermore, containments between these submodules in the exceptional prime cases

are exactly those that are allowed by the dimensions of the module’s composition

factors. In fact, this observation in the case of smaller modules inspired the above

proof of the submodule structure of L3V for the exceptional prime cases with G =

E7(q). Our observation also leads to the following conjecture.

Conjecture 4.6.12. Theorem 4.6.11(iii) holds even if we do not assume that q = p,

and the submodule structure of L3V is equivalent to that of L3W and that of

(L3W )|G even if p is an exceptional prime for G.

Suppose now that p is an exceptional prime for G ∈ E6(q), E7(q), with X the

corresponding group of Lie type defined over Fp, X the simply connected version

of X, and U a minimal Fp[X]-module. We can adapt the proof of the uniserial

cases of Theorem 4.6.11(i) to show that if N is a uniserial submodule of L3U , then

NK is uniserial. Since L3U is a direct sum of uniserial submodules that have no

common composition factors (even when X = E6(5)), it follows from Theorem 2.8.19

that L3U and (L3U)K have equivalent submodule structures. Hence the submodule

lattice of each of L3W , (L3V )K ∼= (L3W )|G and L3V is equivalent to a sublattice of

the submodule lattice of L3U . Moreover, the length of a composition series is the

same for all of these modules by Theorem 4.6.4. In particular, if each dimension of

a uniserial direct summand of L3U is also the dimension of a submodule of L3W ,

then (L3V )K and L3V also contain submodules of these dimensions. In this case,

Theorem 2.8.19 implies that Conjecture 4.6.12 holds.

2Recall from Corollary 4.2.3 that G ∼= G when G = G2(q).

Chapter 5

Overgroups of exceptional

Chevalley groups

5.1 General results

In this chapter, we will determine part of the overgroup structure in GL(d, q) of

the simply connected version G of each exceptional Chevalley group defined over the

field Fq of odd characteristic, where d is the dimension of a minimal Fq[G]-module.

Later in this thesis, we will use this information to determine, in each case, the

stabiliser in GL(d, q) of the proper nonzero G-submodules of the second or third Lie

power of Fdq . This chapter also includes a section describing the action of G2(q) on

the largest proper submodule of the second Lie power of Fdq , which we can identify

with A2(Fdq) by Lemma 2.11.8. Moreover, we define and briefly consider the larger

family of groups G2(F), where F is any field of characteristic other than 2.

All results in this chapter are our own, unless stated otherwise. Additionally,

throughout this chapter and the next, we use standard Atlas [18] notation for group

extensions, as well as the notation described in the following hypothesis. Here, we

use Lemma 4.6.8, which states that each minimal Fq[G]-module is faithful.

Hypothesis 5.1.1. We assume that:

q is a power of an odd prime p;

G is an exceptional Chevalley group defined over Fq;

G is the simply connected version of G, as defined in Lemma 4.2.1;

V is a minimal Fq[G]-module;

d := dim(V );

ZGL := Z(GL(d, q)); and

ZSL := Z(SL(d, q)).

We also consider G as the image in GL(d, q) of the representation afforded by V .

By Lemma 4.6.8, d is given in Table 4.5.22, i.e.,

(G, d) ∈ (G2(q), 7), (F4(q), c), (E6(q), 27), (E7(q), 56), (E8(q), 248),

101

102 Chapter 5. Overgroups of exceptional Chevalley groups

where c = 25 if p = 3 and c = 26 otherwise. Although this lemma applies with

p = 2, we will not consider this case as some of the important results below only

hold when p > 2. Lemma 4.6.8 also shows that all minimal Fq[G]-modules are

absolutely irreducible, with the images of the afforded representations forming a

single conjugacy class in GL(d, q). This means that if a result in this chapter applies

to a particular absolutely irreducible subgroup of GL(d, q) that is isomorphic to G,

then it applies to every such subgroup. In fact, Proposition 4.6.10 shows that we can

choose V to be any faithful d-dimensional Fq[G]-module, with G the image of the

afforded representation. Note also that no conjugate of G in GL(d, q) can be written

over a proper subfield of Fq by Lemma 4.6.8. Finally, G ∼= G/Z(G) by Lemma 4.2.1,

and G is quasisimple by Theorem 4.2.4.

In the remainder of this section, we present general results that we will apply

to the individual families of groups G in this chapter’s subsequent sections. For

the next lemma, which follows from [53, Lemma 2.10.15], recall that each reflexive

bilinear form on V is symmetric or alternating by Theorem 2.5.2. Additionally, by

Definition 2.5.3, if a non-degenerate σ-Hermitian form is defined on V , then q is a

square, and σ is the automorphism of Fq that maps each µ ∈ F to µq1/2

. In this

case, Proposition 4.2.31 implies that there is a corresponding automorphism σ of G

that transforms each matrix element of the group by raising each entry to its q1/2-th

power.

Lemma 5.1.1. The group G preserves a non-degenerate reflexive bilinear form on

V if and only if V and V ∗ are isomorphic as modules. Additionally, G preserves a

non-degenerate σ-Hermitian form on V if and only if q is a square and V ∗ and V σ

are isomorphic as modules.

Proposition 5.1.2. The centre of G is a subgroup of ZGL.

Proof. Clearly, Z(G) is a subgroup of the centraliser CGL(d,q)(G). As G is an

absolutely irreducible subgroup of GL(d, q), this centraliser is equal to ZGL by [13,

p. 38].

The following results discuss containments of G in geometric subgroups and

maximal C9-subgroups of classical groups, as defined in Section 2.10.

Lemma 5.1.3. Let S ∈ SL(d, q), Sp(d, q),Ωε(d, q), where ε ∈ ,+,−. Then no

C2-subgroup of S contains G.

Proof. Suppose that H is a C2-subgroup of S that contains G. Then by Definition

2.10.2, H stabilises a decomposition of V as V = V1 ⊕ · · · ⊕ Vm, where m > 1 is

an integer dividing d, and where Vi is a subspace of V of dimension d/m for each

i ∈ 1, . . . ,m. In particular, H permutes the components of this decomposition,

and hence there exists a homomorphism ρ from H to the symmetric group Sm such

that (H)ρ is the permutation group induced by H on V1, . . . , Vm. Additionally,

5.1. General results 103

the subgroup (G)ρ of Sm is isomorphic to G/ ker ρ|G. Using Table 4.1.3, we see that

the largest power of q that divides |G| does not divide |Sm| for any m 6 d, unless

G = E6(3), in which case the factor (330 − 1) of |G| does not divide |Sm| for any

such m. It follows from Lagrange’s Theorem that |G| does not divide the order of

G/ ker ρ|G. As G ∼= G/Z(G), and as G has no nontrivial proper normal subgroup

other than Z(G) by Proposition 4.6.9, we have ker ρ|G = G and (G)ρ = 1.

Now, we can identify ker ρ with a subgroup of B := GL(V1)× · · · ×GL(Vm) [13,

Ch. 2.2.2]. Let πi be the projection map from B to GL(Vi) for each i ∈ 1, . . . ,m.As G 6 ker ρ, there exists j ∈ 1, . . . ,m such that the quotient (G)πj of G is

nontrivial. Moreover, (G)πj is a subgroup of GL(Vj) ∼= GL(d/m, q). However,

Proposition 4.6.10 shows that no nontrivial quotient of G is a subgroup of GL(n, q)

for any n < d. This is a contradiction, and thus G lies in no C2-subgroup of S.

Lemma 5.1.4. Let S ∈ SL(d, q), Sp(d, q),Ωε(d, q), where ε ∈ ,+,−. Then no

C4-subgroup of S contains G.

Proof. Suppose that H is a C4-subgroup of S that contains G. Then H lies in the

central product K := GL(m, q) GL(n, q), where1 1 < m <√d and mn = d [13,

Ch. 2.2.4]. Bamberg et al. [7, Lemma 5.5] show that K stabilises a subspace of A2V

of dimension(m2

)(n+1

2

), and so the subgroup G of K also stabilises this subspace.

However, Theorem 4.6.11 shows that A2V does not contain a G-submodule of di-

mension(m2

)(n+1

2

), for any permitted values of m and n. This is a contradiction,

and thus G lies in no C4-subgroup of S.

We now consider the geometric subgroups of the remaining classes.

Lemma 5.1.5. Let S ∈ SL(d, q), Sp(d, q),Ωε(d, q), where ε ∈ ,+,−. If G

preserves a non-degenerate σ-Hermitian or reflexive bilinear form β on V , then the

largest group of similarities of β in SL(d, q) is the only geometric subgroup of SL(d, q)

that contains G. Moreover, this group of similarities is a maximal C8-subgroup of

SL(d, q). If G does not preserve such a form on V , or if S 6= SL(d, q), then G does

not lie in any geometric subgroup of S.

Proof. Note that since the subgroup G of GL(d, q) is perfect, it lies in the derived

subgroup of GL(d, q), which is SL(d, q) by Proposition 2.6.2. We have from Lemmas

5.1.3 and 5.1.4 that G does not lie in any C2- or C4-subgroups of S. Furthermore, as

G is absolutely irreducible and no conjugate of G in GL(d, q) can be written over a

proper subfield of Fq, Proposition 2.10.4 implies that it does not lie in any subgroup

of S in C1 ∪ C3 ∪ C5.

Definition 2.10.2 shows that C6-subgroups of S are only defined when d is a power

of a prime, and that C7-subgroups of S are only defined when d is a power of a positive

1The inequality m <√d holds because, in the cases under consideration,

√d is not an integer

when d is even.


integer less than d. Thus, in the cases we are considering, C6-subgroups are only

defined when d ∈ 7, 25, 27, and C7-subgroups are only defined when d ∈ 25, 27.If H6 is a C6-subgroup of S with d ∈ 7, 25, 27, then S = SL(d, q) and H6 has shape

(A r1+m+ ).Sp(m, r), where |A| 6 27 and (d, r,m) ∈ (7, 7, 2), (25, 5, 4), (27, 3, 6),

and where r1+m+ denotes the extraspecial group of order r1+m and exponent r [13,

Table 2.9]. Additionally, if H7 is a C7-subgroup of S with d ∈ 25, 27, then the

shape of H7 is either B.PSL(n, q)t.C.St or Ω(n, q)t.D.St, where |B| 6 5, |C| 6 81,

|D| 6 4 and (d, n, t) ∈ (25, 5, 2), (27, 3, 3) [13, Table 2.10]. By considering group

orders, using the tables in Section 4.1 and the data in [13, Ch. 1.6.4], we see that,

in each case, |H6| or |H7| is not divisible by the order of the corresponding group

G for any odd q. Since |G| divides |G|, it follows from Lagrange’s Theorem that G

does not lie in H6 or in H7, in the respective cases.

We have shown that G does not lie in any subgroup of S in C1 ∪ · · · ∪ C7. Thus,

by Definition 2.10.2, if G lies in a geometric subgroup H of S, then S = SL(d, q)

and H is a C8-subgroup of S, i.e., H is the full group of similarities in S of a non-

degenerate σ-Hermitian or reflexive bilinear form. As G is perfect and absolutely

irreducible, Lemma 2.9.14 implies that G preserves this form absolutely. In fact,

since no conjugate of G in GL(d, q) can be written over a proper subfield of Fq,Lemma 2.9.14 shows that (up to scalar multiplication) this form is β. In other

words, H is the full group of similarities of β in S. As S does not preserve β by

Proposition 2.5.14, H is a proper subgroup of S. In fact, H is maximal in S, as

shown by Bray, Holt and Roney-Dougal [13, Table 8.35] for d = 7, and by Kleidman

and Liebeck [53, Proposition 7.8.1, Lemma 8.1.6] for the other values of d under

consideration.

Recall now that for a subgroup X of GL(d, q), X = ZGLX/ZGL.

Theorem 5.1.6. Let S ∈ SL(d, q), Sp(d, q),Ωε(d, q), where ε ∈ ,+,−. In

addition, let H be a maximal C9-subgroup of S that contains G, and assume that

H∞/Z(H∞) is isomorphic to a simple group of Lie type defined over a field of

characteristic p. Then H = NS(G). Moreover, if G preserves a non-degenerate

σ-Hermitian or reflexive bilinear form on V , then S 6= SL(d, q).

Proof. As G is perfect, it is contained in H∞, which is quasisimple by Lemma

2.10.6 and absolutely irreducible by Definition 2.10.3. Suppose that H∞/Z(H∞) is

isomorphic to a simple group of Lie type X defined over a field of characteristic p.

We have from Lemma 2.10.6 that X ∼= H∞/Z(H∞) is isomorphic to H∞, and that

PGL(d, q) contains a copy of X. Furthermore, we have from Proposition 5.1.2 that

G ∼= G/Z(G) = G/(G∩ZGL), which is isomorphic to G by the Second Isomorphism

Theorem. Since G 6 H∞, the Correspondence Theorem implies that X contains a

copy of G.


Now, suppose that m > 1 is an integer such that X is a subgroup of PGL(m, q),

with X ∼= Q for some absolutely irreducible group Q 6 GL(m, q) that cannot

be written over a proper subfield of Fq. Then the simply connected version X of

X, as defined in Lemma 4.2.1, has an absolutely irreducible Fq-representation of

dimension m that cannot be written over a proper subfield of Fq [56, p. 135–136].

Since X ∼= H∞, with H∞ an absolutely irreducible group that cannot be written

over a proper subfield of Fq by Definition 2.10.3, we require that d is an allowed

value for m. Since p > 2, Table 4.1.2 implies that if X = tY`(r), with r a power

of p, then (t, Y ) /∈ (2, B), (2, F ). Furthermore, if d = xy for integers x > 0 and

y > 1, then the tuple (d, x, y) lies in the set (25, 5, 2), (27, 3, 3). Hence Proposition

4.6.6 implies that either X = 2G2(r) or r ∈ q, q1/t, q2, q2/t, q3, q3/t. In particular,

if X 6= 2G2(r), then r ∈ q2, q2/t is only possible if G = F4(q) with p = 3, and

r ∈ q3, q3/t is only possible if G = E6(q).

As X contains a copy of G, we have from Lagrange’s Theorem that |G| divides

|X|. In particular, each factor of |G| of the form (qi − 1), with i an integer, must

divide |X|. It follows from the tables in Section 4.1 that if X is a classical group,

then a lower bound j for the dimension of the general linear group associated with

X is given by

(G, j) ∈ (G2(q), 6), (F4(q), s), (E6(q), 8), (E7(q), 18), (E8(q), 30),

where s = 6 if p = 3 and s = 12 otherwise. For example, if G = G2(q) then

|PSL(5, r)| is not divisible by |G|. In fact, since d must be an allowed value for m, it

follows from Theorem 4.6.4 and Lubeck’s [56] lists of irreducible modules for linear

algebraic groups that we can improve the lower bound for G = G2(q) to j = 7, and

the lower bound for each of G = F4(q) (for all odd p) and G = E7(q) to j = 19.

Suppose that X is a classical group, where the dimension of the associated general

linear group is an integer b > j. Kleidman and Liebeck [53, Proposition 5.4.11] state

that the integer m defined above satisfies m > b, and if m > b, then m > n, where

(X, n) is in the set(PSL(b, r),

(b

2

));

(PSU(b, r),

(b

2

));

(PSp(b, r),

(b

2

)− k)

;

(Ω(b, r), 2(b−1)/2), b 6 13;

(Ω(b, r),

(b

2

)), b > 15;

(PΩ±(b, r), 2b/2−1), b 6 12;

(PΩ±(b, r),

(b

2

)), b > 14

,

with k ∈ 1, 2 depending on b and p. Observe that, if G 6= E6(q), then in order for d

to be an allowed value form, we require b = d. If G = E6(q) and b 6= d, then Lubeck’s

[56] lists of irreducible modules for linear algebraic groups show that X = PSp(8, r).

In this case, Lubeck shows that there is no irreducible 3-dimensional module for the

linear algebraic group X associated with X, and hence r = q by Proposition 4.6.6.


We see from the tables in Section 4.1 that PSp(8, q) is too small to contain E6(q),

and hence we have b = d even when G = E6(q). Furthermore, since m cannot equal

2 or 3 for any classical candidate for X, we have r ∈ q, q1/t. If X = PSU(d, r),

then the minimal Fq[X]-module corresponds to the irreducible Fp[X]-module L(aλi),

where a is a power of p and i ∈ 1, d− 1, and if X = PΩ−(d, r), then the minimal

Fq[X]-module corresponds to L(aλ1) [55, §1] (see also [56, Appendix A.3]). If γ is

the graph automorphism of X of order 2, then γ interchanges L(aλ1) and L(aλd−1)

in the former case and fixes L(aλ1) in the latter [53, p. 180, p. 192]. We therefore

have r = q1/2 in the former case and r = q in the latter, by Proposition 4.6.6. In all

other cases, t = 1, and hence r = q. Thus if X is a classical group, then

X ∈ PSL(d, q),PSU(d, q1/2),PSp(d, q),Ω(d, q),PΩ±(d, q).

If Q 6 GL(d, q) is the image of an irreducible representation such that Q ∼=X and ZGL 6 Q, then this representation is quasi-equivalent to the irreducible

representation with image ZGLH∞ [53, Proposition 5.4.11]. It follows from Lemma

2.8.11 that ZGLH∞ is conjugate in GL(d, q) to ZGLR, where

R ∈ SL(d, q), SU(d, q1/2), Sp(d, q),Ω(d, q),Ω±(d, q).

As R is perfect by Proposition 2.6.8, there exists x ∈ GL(d, q) such that

H∞ = (H∞)′ = (ZGLH∞)′ = ((ZGLR)x)′ = (ZGLR

x)′ = (Rx)′ = (R′)x = Rx,

where the equality (Rx)′ = (R′)x follows from Proposition 2.1.5. Lemma 2.9.14

implies that H∞ preserves at most one nonzero bilinear or σ-Hermitian form, while

we require from Definition 2.10.3 that H∞ preserves exactly the σ-Hermitian and

reflexive bilinear forms preserved by S. It follows from Proposition 2.5.13 and the

definitions of the possible candidates forR thatH∞ = S. However, asH∞ 6 H < S,

this is a contradiction. Therefore, X is not a classical group.

We have shown that X is an exceptional group of Lie type. As d is an allowed

value for m, Lubeck’s [56] lists of irreducible modules for linear algebraic groups

imply that if G = Y ′`′(q) and X = tY`(r), then either Y ′`′ = Y`, or

(Y ′`′ , Y`) ∈ (F4, G2), (E6, G2), (E7, D4), (E8, G2).

In particular, if X = 2G2(r), then p = 3 by Table 4.1.2, and hence d ∈ 7, 27 [56,

Appendix A.49], i.e., d < 72. Thus in this case, r = q by Proposition 4.6.7. In each

other case, there is no irreducible module for the linear algebraic group X of dimen-

sion 2 or 3 [56], and hence r ∈ q, q1/t. In particular, if G = E6(q) and X = 2E6(r),

then Proposition 4.2.5, Lemma 4.2.32 and Table 4.5.22 imply that the absolutely

irreducible d-dimensional X-module corresponds to the irreducible Fp[E6]-module

L(aλi), where a is a power of p and i ∈ 1, 6. The graph automorphism of E6

of order 2 interchanges L(aλ1) and L(aλ6) [53, p. 180, p. 192], and hence r = q1/2


by Proposition 4.6.6. Since |G| must divide |X|, it follows from Table 4.1.3 that

(t, Y, `) = (1, Y ′, `′), and hence r = q and X ∼= G.

Now, we have from above that

ZGLG/ZGL = G ∼= G ∼= X ∼= H∞ = ZGLH∞/ZGL.

As G 6 H∞, the Correspondence Theorem implies that ZGLG = ZGLH∞. Since G

and H∞ are perfect, it follows that

G = G′ = (ZGLG)′ = (ZGLH∞)′ = (H∞)′ = H∞.

The group H∞ is normal in H, and therefore H 6 NS(G). Furthermore, since

G < S, the Correspondence Theorem implies that ZGLS normalises ZGLG if and only

if S normalises its proper subgroup G ∼= G. The simple group S does not normalise

G, and hence ZGLS does not normalise ZGLG. However, ZGL does normalise ZGLG,

which means that S does not. Since S normalises ZGL, this means that S does not

normalise G. Thus NS(G) is a proper subgroup of S, and so H = NS(G). Finally,

if H∞ = G preserves a non-degenerate bilinear or σ-Hermitian form on V , then

Definition 2.10.3 shows that H can only be a C9-subgroup of S if S 6= SL(d, q).

Note that the maximal C9-subgroups of SL(7, q) and Ω(7, q) have been deter-

mined by Bray, Holt and Roney-Dougal [13, Table 8.36, Table 8.40]. Therefore,

although we have included the case G = G2(q) in the above theorem and its proof

for completeness, we will not apply the theorem in this case.

For the following lemma, recall from Proposition 2.8.8 that we can twist V by

an automorphism α of G to obtain the G-module V α.

Lemma 5.1.7. Let N := NGL(d,q)(G). Then N/(ZGLG) is isomorphic to the sub-

group Inn(G)α | α ∈ Aut(G), V α ∼= V of Out(G).

Proof. First, note that N normalises both ZGL and G, and thus ZGLG C6 N . If

x ∈ N , then conjugation by x is an automorphism of N with Gx = G, and so x

induces an automorphism of G. In particular, this defines an action θ : N → Aut(G),

and it is clear that ker θ = CN(G) = CGL(d,q)(G)∩N . As G is absolutely irreducible,

we have CGL(d,q)(G) = ZGL by [13, p. 38], and thus ker θ = ZGL. Observe that θ|Gis an epimorphism from G to Inn(G), and it follows that ZGLG is the full preimage

of Inn(G) under θ. Hence the map θ′ : N/(ZGLG) → Aut(G)/Inn(G) = Out(G)

defined by (ZGLGx)θ′ := Inn(G)(x)θ for all x ∈ N is a well-defined homomorphism.

In fact, θ′ is a monomorphism, as ker θ′ = ZGLG.

Now, if G preserves a non-degenerate bilinear or σ-Hermitian form β on V , then

let C be the largest group of similarities of β in GL(d, q). In this case, since G

is absolutely irreducible, Lemma 2.9.14 implies that N = NC(G). If G does not

preserve such a form on V , then let C := GL(d, q), in which case N = NC(G)

by definition. Bray, Holt and Roney-Dougal [13, Lemma 4.4.3] show that, since V


is a faithful, absolutely irreducible module for the quasisimple group G, an outer

automorphism α of G is induced by an element x ∈ NC(G) = N if and only if

V α ∼= V . Thus N/(ZGLG) ∼= Inn(G)α | α ∈ Aut(G), V α ∼= V . Note that

this subset of Out(G) is well-defined, as V β ∼= V for each β ∈ Inn(G) [13, p. 39].

Moreover, this subset is indeed a subgroup of Out(G), as if V α ∼= V and V β ∼= V

for α, β ∈ Aut(G), then V αβ ∼= V and V α−1 ∼= V .

Proposition 5.1.8. Let S ∈ Sp(d, q),Ωε(d, q), and let C be the largest group of

similarities in GL(d, q) of the non-degenerate form preserved by S. Additionally,

suppose that:

(i) H is a perfect subgroup of S such that NGL(d,q)(H) = (ZGLH).B, where B is

a soluble group;

(ii) NS(H) is the unique maximal subgroup of S containing H; and

(iii) M := C ∩ SL(d, q) is the unique maximal subgroup of SL(d, q) containing H.

Then a subgroup K of SL(d, q) containing H also normalises H if and only if S 66 K.

Additionally, if B = 1, then NSL(d,q)(H) = Z(SL(d, q))H.

Proof. Let K be a subgroup of SL(d, q) that contains H, but not S. Then K ∩ S is

a subgroup of NS(H) by (ii), and this is a subgroup of (ZGLH).B by (i). It follows

that H∞ 6 (K ∩ S)∞ 6 ((ZGLH).B)∞. As B ∼= ((ZGLH).B)/(ZGLH) is soluble,

((ZGLH).B)∞ = (ZGLH)∞. This is equal to H∞ by Proposition 2.1.4. Thus the

perfect group H = H∞ is the characteristic subgroup (K∩S)∞ of K∩S. Now, K is a

proper subgroup of SL(d, q), and so K 6M by (iii). Additionally, S C6 C [53, p. 14],

and so S C6 M , which gives K ∩ S C6 K. Therefore, the characteristic subgroup H

of K ∩ S is normal in K, as required. Conversely, NSL(d,q)(H) ∩ S = NS(H) < S,

and thus no subgroup of NSL(d,q)(H) contains S. If B = 1, then NSL(d,q)(H) =

(ZGLH) ∩ SL(d, q), which is equal to Z(SL(d, q))H by Dedekind’s Identity.

5.2 G2(F) and the octonion algebra

Recall from Corollary 4.2.3 that G ∼= G when G = G2(q). In Theorem 4.6.11,

we determined that the exterior square of each irreducible 7-dimensional Fq[G2(q)]-

module contains a unique maximal 14-dimensional submodule. In this section, we

construct a basis for this submodule, which will allow us to determine exactly how

G2(q) acts on this submodule. Our approach also yields some results in the more

general case of the group G2(F), which we will define shortly, where F is any field

of characteristic other than 2.

We now describe a nonassociative algebra that will allow us to define G2(F). The

following definition and the subsequent two paragraphs are based on information

from [73, p. 119–122].

5.2. G2(F) and the octonion algebra 109

Definition 5.2.1. Let W be the 8-dimensional vector space over a field F of charac-

teristic other than 2, with basis vectors denoted 1, i0, i1, . . . , i6. Then W becomes

the octonion algebra O when it is equipped with the following multiplication rules

that extend linearly:

(i) 1e = e1 = e for each e ∈ 1, i0, i1, . . . , i6;

(ii) i2t = −1 for each t ∈ Z7; and

(iii) itit+1 = it+3, it+1it+3 = it, and it+3it = it+1 for each t ∈ Z7, with subscripts

interpreted modulo 7.

It follows that 1 is the unique multiplicative identity of O, and that itis = −isitfor all distinct s, t ∈ Z7. The elements of O are called octonions, and the real and

imaginary octonions are the elements of 〈1〉 and of O := 〈i0, i1, . . . , i6〉, respec-

tively. The real part of an octonion x = b1 +∑6

t=0 atit, with b, a0, a1, . . . , a6 ∈ F,

is Re(x) := b1; its imaginary part2 is Im(x) :=∑6

t=0 atit ∈ O; and its conjugate is

x := b1−∑6

t=0 atit. The functions Re and Im are F-linear projection maps from

O = 〈1〉 ⊕O to 〈1〉 and O, respectively. A useful fact is that if x ∈ O, i.e., if b = 0,

then:

Im(x2) = Im(6∑s=0

6∑t=0

asatisit) = Im(6∑

s,t=0s<t

asat(isit + itis) +6∑t=0

a2t i

2t )

= Im(6∑t=0

−a2t1) = 0.

(5.2.1)

We also have xx ∈ 〈1〉 for all x ∈ O. If we identify 〈1〉 with F, then we can

define a symmetric bilinear form on O by (x, y)β := Re(xy) for x, y ∈ O. Finally,

we denote the group of automorphisms of O by G2(F). Explicitly, G2(F) is the group

of bijections from O to itself that preserve vector addition, scalar multiplication and

octonion multiplication. Elements of G2(F) also preserve the symmetric bilinear

form β. If F = Fq (with q odd), then G2(F) is the simple group of Lie type G2(q).

The following proposition and its proof are from Wilson [73, p. 119, p. 121].

Proposition 5.2.2. The group G2(F) stabilises O (as a subspace of the algebra O).

Proof. Let g ∈ G2(F), and let x ∈ O. Since g is an automorphism of O, we have

xg = (1x)g = 1gxg. We similarly obtain xg = xg1g, and hence 1g = 1. Thus for

each a ∈ F, (a1)g = a(1)g = a1, and hence G2(F) stabilises 〈1〉. It follows from

Proposition 2.5.9 that G2(F) stabilises the orthogonal complement 〈1〉⊥ of 〈1〉 with

respect to β. For a ∈ F and y ∈ O, we have

(a1, y)β = Re(a1y) = Re(ay) = aRe(y) = aRe(y).2Note that, unlike the real and imaginary parts of a complex number, the real and imaginary

parts of x ∈ O are vectors, not scalars.


Definition 2.5.5 implies that y ∈ 〈1〉⊥ if and only if (a1, y)β = 0, which is the case

if and only if y ∈ O. Therefore, G2(F) stabilises O.

Now, let B := is ∧ it | s, t ∈ Z7, s < t. We have from Lemma 2.7.3 that B is a

basis for A2O, and that dim(A2O) =(

dim(O)2

)=(

72

)= 21.

Proposition 5.2.3. There exists a unique linear map f : A2O → O defined by

(x ∧ y)f := Im(xy), and this map is an F[G2(F)]-homomorphism.

Proof. First, let f : O2 → O be the map defined by (x, y)f := Im(xy) for x, y ∈ O.

It follows from Definition 2.7.6 and the linearity of Im that f is bilinear, and f is

alternating by (5.2.1). Thus f satisfies the hypotheses of Proposition 2.7.4, and so

there exists a unique linear map f : A2O → O such that (x ∧ y)f = (x, y)f for all

x, y ∈ O. In particular, for distinct basis vectors is and it, we have (is ∧ it)f = isit,

since this product is imaginary.

Now, O is an F[G2(F)]-module by Proposition 5.2.2, and hence A2O is also an

F[G2(F)]-module by Proposition 2.8.22. The definition of multiplication in O implies

that for all distinct s, t ∈ Z7, we have isit = ±ir for some r ∈ Z7. Since ±ir ∈ O,

we have (±ir)g ∈ O for each g ∈ G2(F) by Proposition 5.2.2. Furthermore, the

definition of G2(F) gives igsigt = (isit)

g = (±ir)g, and hence Im(igsigt ) = igsi

gt . By the

definition of the action of G2(F) on A2O, we therefore have

((is ∧ it)f)g = (isit)g = igsi

gt = (igs ∧ i

gt )f = ((is ∧ it)g)f .

Since g acts linearly on A2O, whose basis B consists of the vectors is∧ it with s < t,

and since f is linear, it follows that ((w)f)g = (wg)f for each w ∈ A2O. Therefore,

the linear map f is an F[G2(F)]-homomorphism.

Lemma 5.2.4. The group G2(F) stabilises the subspace ker f of A2O (as a vector

space), which has dimension 14.

Proof. Since f is an F[G]-homomorphism by Proposition 5.2.3, G2(F) stabilises

ker f . Moreover, it follows from the definition of multiplication in O that for each

t ∈ Z7, we have (it+1 ∧ it+3)f = Im(it) = it. The image of f is therefore equal to O,

with dimension 7. We also have dim(A2O) = 21, and hence the dimension of ker f

is 21− 7 = 14, by the Rank-Nullity Theorem.

We have shown that when char(F) 6= 2, G2(F) stabilises a 14-dimensional sub-

space of A2O corresponding to the alternating form given by (x, y)f = Im(xy) for

x, y ∈ O. Gow [34] proves a stronger result when F also satisfies the condition that

xx 6= 0 for all nonzero x ∈ O. Namely, in this case, for each alternating form on O,

G2(F) stabilises a corresponding 14-dimensional subspace of A2O.

5.2. G2(F) and the octonion algebra 111

Proposition 5.2.5. A basis for the subspace ker f of A2O is

i0 ∧ i1 + i2 ∧ i5, i0 ∧ i1 − i4 ∧ i6, i0 ∧ i2 − i1 ∧ i5, i0 ∧ i2 − i3 ∧ i4,

i0 ∧ i3 + i2 ∧ i4, i0 ∧ i3 + i5 ∧ i6, i0 ∧ i4 + i1 ∧ i6, i0 ∧ i4 − i2 ∧ i3,

i0 ∧ i5 + i1 ∧ i2, i0 ∧ i5 − i3 ∧ i6, i0 ∧ i6 − i1 ∧ i4, i0 ∧ i6 + i3 ∧ i5,

i1 ∧ i3 − i2 ∧ i6, i1 ∧ i3 − i4 ∧ i5.

Proof. Let S be the set of vectors that we have claimed is a basis for ker f . Each

term of each vector in S is a vector in B (possibly with a different sign), and no

scalar multiple of the second term of a given vector in S appears as any other term

in a vector in S. Thus S is a linearly independent set. Furthermore, |S| = 14, which

is equal to dim(ker f) by Lemma 5.2.4. Finally, the linearity and definition of f

from Proposition 5.2.3 and the definition of octonion multiplication from Definition

5.2.1 show that each vector in S is a vector in ker f . For example,

(i0 ∧ i1 + i2 ∧ i5)f = (i0 ∧ i1)f + (i2 ∧ i5)f = Im(i0i1) + Im(i2i5) = i3 + (−i3) = 0.

Therefore, S is a basis for ker f .

Now, the group G2(Fq) = G2(q) is simple and does not act trivially on O. Thus

Proposition 2.8.4 implies that O is a faithful F[G2(q)]-module. The following result

is therefore a consequence of Proposition 4.6.10, where we have used the fact that

G ∼= G when G = G2(q).

Proposition 5.2.6. The space O of imaginary octonions over Fq is an irreducible

Fq[G2(q)]-module.

For a corresponding result in the case where F is an infinite field, see [47, Propo-

sition 4].

Proposition 5.2.7 ([73, p. 121–122]). Each element g ∈ G2(F) is uniquely deter-

mined by the tuple (x, y, z) := (i0, i1, i2)g. Moreover, such a g exists for all imaginary

octonions x, y and z with xx = yy = zz = 1 and (x, y)β = (x, z)β = (y, z)β =

(xy, z)β = 0.

Note that since i0, i1, . . . , i6 is a basis for O, with each octonion in this basis

equal to a finite product of octonions in i0, i1, i2, the image of any imaginary

octonion under g ∈ G2(F) can be determined by (i0, i1, i2)g. Thus this proposition

describes the action of G2(F) on O.

We have from Proposition 5.2.5 that each vector v ∈ ker f can be written as a

linear combination of vectors of the form ia ∧ ib ± ic ∧ id, with a, b, c, d ∈ Z7. The

definition of the action of G2(F) on A2O implies that, for each g ∈ G2(F), vg is

the corresponding linear combination of vectors of the form iga ∧ igb ± igc ∧ i

gd. Thus

we can derive the action of G2(F) on ker f from the action of G2(F) on O given by

Proposition 5.2.7.


Suppose that F = Fq, and let G = G2(q). Since G ∼= G, the module V from Hy-

pothesis 5.1.1 is an irreducible 7-dimensional Fq[G]-module. We can now determine

the action of G on the unique 14-dimensional submodule of A2V from Theorem

4.6.11. Since O is also an irreducible 7-dimensional Fq[G]-module by Proposition

5.2.6, Lemma 4.6.8 gives that V ∼= Oα for some α ∈ Aut(G). Hence A2V ∼= A2(Oα),

which is equal as a module to (A2O)α by Proposition 2.8.23. Corollary 2.8.9 therefore

implies that the 14-dimensional submodule of A2V is U := (ker f)α. The definition

of the action of a group on a module twisted by a group automorphism implies that

for each g ∈ G and each u ∈ U , the image ug associated with the module U is equal

to the image ugα−1

associated with the module ker f .

5.3 Overgroups of G2(q) in GL(7, q)

In this section, we establish results about certain overgroups of G in GL(d, q),

with G = G2(q) and d = 7. Recall from Corollary 4.2.3 that G ∼= G. As the

7-dimensional Fq[G]-module O is irreducible by Proposition 5.2.6, it follows from

Lemma 4.6.8 that G is conjugate in GL(7, q) to the group G2(q) from Section 5.2,

and that we can identify V with O, up to twisting by an automorphism of G, with

V ∼= O as vector spaces. We also know that G2(q) preserves a nonzero orthogonal

form on O, which is non-degenerate by Proposition 2.8.3. Thus Proposition 2.5.13

implies that G preserves a non-degenerate orthogonal form β on V . In this section,

GO(7, q) denotes the general orthogonal group that preserves β, while SO(7, q) and

Ω(7, q) denote the associated special orthogonal group and generally quasisimple

classical group, respectively.

Proposition 5.3.1 ([13, Table 8.40]). The group G is a maximal C9-subgroup of

Ω(7, q).

Indeed, since the perfect group G preserves β, it lies in (GO(7, q))∞, which is

equal to Ω(7, q) by Proposition 2.6.10. Furthermore, since G is a non-abelian simple

group, Definition 2.10.3 and Lemma 4.6.8 imply that G is a C9-subgroup of Ω(7, q).

Lemma 5.3.2. The normaliser N of G in GL(7, q) is equal to ZGLG. Moreover, N

is not maximal in GL(7, q).

Proof. The cosets of Inn(G) in Aut(G) are defined by the field automorphisms of

G (i.e., the coset defined by the field automorphism φ is Inn(G)φ), and also by the

graph automorphism of G when p = 3 [73, p. 128]. Lemma 4.6.8 shows that if α

is a nontrivial field or graph automorphism of G, then the Fq[G]-module V is not

isomorphic to V α. It follows from Lemma 5.1.7 that N = ZGLG. Furthermore,

G < Ω(7, q) by Proposition 5.3.1. By considering group orders [13, Ch. 1.6.4], we

have N < ZGLΩ(7, q) < GL(7, q), and therefore N is not maximal in GL(7, q).

5.4. Overgroups of F4(q) in GL(25, q) or GL(26, q) and of E8(q) in GL(248, q) 113

GL(7, q)

Z(GL(7, q))Ω(7, q)

Z(GL(7, q))GNGL(7,q)(G) =

Z(SL(7, q))GNSL(7,q)(G) =

SL(7, q)

Z(SL(7, q))SO(7, q)

SO(7, q)

Ω(7, q)

G

Figure 5.3.1: Some overgroups of G = G2(q) in GL(7, q), for an odd prime power q.Double edges indicate maximal containment, and subgroups connected by dashededges are equal when q 6≡ 1 (mod 7).

Lemma 5.3.3. The only maximal subgroup of SL(7, q) that contains G is the full

group of similarities of β in SL(7, q), and this subgroup is equal to ZSLSO(7, q).

Proof. As G preserves the non-degenerate orthogonal form β, Lemma 5.1.5 implies

that the only maximal geometric subgroup of SL(7, q) that contains G is the full

group of similarities of β in SL(7, q). Additionally, this group of similarities is equal

to ZSLSO(7, q) [13, Table 8.35]. By Theorem 2.10.5, any other maximal subgroup

of SL(7, q) that contains G is a maximal C9-subgroup. However, each maximal C9-

subgroup of SL(7, q) is isomorphic to ZSL×PSU(3, 3) [13, Table 8.36], which cannot

contain G by the tables in Section 4.1 and Lagrange’s Theorem.

Now, since G is a maximal subgroup of Ω(7, q), which is simple by Proposition

2.6.9, we have NΩ(7,q)(G) = G. Therefore, if we set H = G and S = Ω(7, q), then

the above results show that the hypotheses of Proposition 5.1.8 are satisfied, with

B = 1 and M = ZSLSO(7, q). Proposition 5.1.8 therefore yields the following.

Proposition 5.3.4. The subgroups of SL(7, q) that contain G but do not con-

tain Ω(7, q) are the subgroups K such that G 6 K 6 NSL(7,q)(G). Additionally,

NSL(7,q)(G) = ZSLG.

Figure 5.3.1 summarises what we have proved about the overgroups of G in

GL(7, q). Recall from Definition 2.6.7 that Ω(7, q) is maximal in SO(7, q). Note

also that |ZSL| = (q − 1, 7) by Proposition 2.6.1. Hence if H ∈ SO(7, q), G, then

ZSLH = H when q 6≡ 1 (mod 7). Furthermore, since SO(7, q) ∩ ZSL = 1 [13, Table

1.3], Lagrange’s Theorem implies that H is maximal in ZSLH when q ≡ 1 (mod 7).

5.4 Overgroups of F4(q) in GL(25, q) or GL(26, q) and of E8(q)

in GL(248, q)

In this section, we examine the overgroups of G in GL(d, q), where (G, d) ∈(F4(q), c), (E8(q), 248), with c = 25 if p = 3 and c = 26 otherwise. The groups


F4(q) and E8(q) possess some similar properties, and hence we are able to consider

them together. Note that Corollary 4.2.3 implies that, in each case, G ∼= G. We

also explore in this section how F4(q) can be described as the automorphism group

of a certain algebra. First, we present three necessary number theoretic results, the

second of which is well-known. Here, if r and r′ are primes, then we say that r is a

square in Fr′ if (r mod r′) is a square in Fr′ .

Proposition 5.4.1 ([13, Proposition 1.13.8]). Let r and r′ be odd primes. Then r

is a nonzero square in Fr′ if and only if either (r − 1)(r′ − 1)/4 is even and r′ is a

nonzero square in Fr, or (r − 1)(r′ − 1)/4 is odd and r′ is not a square in Fr.

Proposition 5.4.2. Let r be a prime, let n > 2 be an integer, and let a ∈ F×r . If

either r or n is even, then a is a square in Frn . However, if both r and n are odd,

then a is a square in Frn if and only if a is a square in Fr.

Proof. Let s := (1 + r + r2 + · · · + rn−1). It is well-known that the multiplicative

group F×rn is cyclic of order rn − 1 = s(r − 1). Let x be a generator for this group.

Since Fr ⊂ Frn , the Fundamental Theorem of Cyclic Groups implies that F×r is the

unique (cyclic) subgroup of F×rn of order r − 1. It follows that xs is a generator for

F×r , and that a = (xs)i = xis for some positive integer i.

If r = 2, then a = 1, which is clearly a square in Frn . Therefore, suppose that r

is odd. If n is even, then s is even, and thus b := xis/2 ∈ Frn and b2 = a, i.e., a is a

square in Frn . Finally, we assume that n is odd. If a is a square in Frn , then there

must exist a nonnegative integer j < rn − 1 such that (xj)2 = xis, i.e., x2j = xis,

i.e., 2j ≡ is (mod rn − 1). Since 2j and rn− 1 are both even, is must also be even.

Note that s is odd, which means that i is even. Hence b := (xs)i/2 ∈ Fr and b2 = a,

and thus a is a square in Fr. Conversely, if a is a square in Fr, then a = k2 for some

k ∈ Fr. Here, we also have k ∈ Frn , and therefore a is a square in Frn .

Proposition 5.4.3. Suppose that p > 3. Then −1 is a square in Fq if and only if

q ≡ 1, 5 (mod 12), and 3 is a square in Fq if and only if q ≡ 1, 11 (mod 12).

Proof. We first assume that q = p. It is well-known and easy to check that −1 is a

square in Fp if and only if p ≡ 1 (mod 4), i.e., if and only if p ≡ 1, 5 (mod 12) (since

no prime is congruent to 9 modulo 12). However, in order to determine when 3 is a

nonzero square in Fp, we must use Proposition 5.4.1. Observe that (3− 1)(p− 1)/4

is even if and only if p ≡ 1 (mod 4), and p is a nonzero square in F3 if and only if

p ≡ 1 (mod 3). Thus 3 is a nonzero square in Fp if and only if p ≡ 1, 11 (mod 12).

Now, let q be any power of p. As p > 3 is prime, we have p ≡ 1, 5, 7, 11 (mod 12).

Additionally, for each positive integer n, pn mod 12 = (p mod 12)n mod 12. In each

case, pn ≡ 1 (mod 12) when n is even, and hence pn ≡ p (mod 12) when n is odd.

It follows from Proposition 5.4.2 that −1 is a square in Fq if and only if q ≡ 1, 5

(mod 12), and that 3 is a nonzero square in Fq if and only if q ≡ 1, 11 (mod 12).


Now, let O be the octonion algebra over Fq, as defined in Definition 5.2.1.

Definition 5.4.4 ([73, p. 149]). A (square) matrix A defined over O is Hermitian

if AT = A, where A is the matrix obtained by conjugating each entry in A.

Definition 5.4.5 ([73, p. 148–149]). Let W be the 27-dimensional space of 3 × 3

Hermitian matrices defined over O, i.e., the set of matrices

α1 β3 β2

β3 α2 β1

β2 β1 α3

with

β1, β2, β3 ∈ O and α1, α2, α3 ∈ 〈1〉 ⊂ O. The non-associative Albert algebra A over

Fq is W equipped with the multiplication defined by A B := 12(AB + BA) for

A,B ∈ W , where AB denotes the standard matrix product of A and B.

Theorem 5.4.6 ([73, p. 149–150, p. 156]). The exceptional group of Lie type F4(q)

is the automorphism group of the Albert algebra A. If p > 3, then F4(q) preserves

the orthogonal form α on A, where (A,B)α is the trace of A B for A,B ∈ A,

and F4(q) acts irreducibly on the 26-dimensional subspace A of trace 0 matrices.

If p = 3, then F4(q) acts irreducibly on the 25-dimensional quotient space A/〈I〉,where I is the 3× 3 identity matrix over Fq.

The Albert algebra was first studied in detail by Albert [2] in 1934, but over Rinstead of a finite field. Since F4(q) is the automorphism group of the algebra A, it

acts linearly on this space, which means that F4(q) does indeed act linearly on the

stabilised 26-dimensional subspace A when p > 3, or on the 25-dimensional quotient

space A/〈I〉 when p = 3.

Lemma 5.4.7. Suppose that p > 3. Then F4(q) is a subgroup of Ωε(26, q), where

ε :=

+, if q ≡ 1, 7 (mod 12),

−, if q ≡ 5, 11 (mod 12).

Proof. Let β be the restriction of the orthogonal form α to A. As F4(q) preserves

β, Definition 2.6.6 implies that, as long as β is non-degenerate, F4(q) 6 GOε(26, q),

with ε ∈ +,−. In fact, if this is the case, then since F4(q) is perfect, we have

F4(q) = (F4(q))∞ 6 (GOε(26, q))∞ = Ωε(26, q), where the last equality is from

Proposition 2.6.10. Observe that if A is the 26-dimensional anti-diagonal matrix

with each anti-diagonal entry equal to 1, then det(A) ≡ −1 (mod p). Thus, by

Definition 2.5.12, F4(q) 6 Ω+(26, q) if det(Jβ) and −1 are both squares or both

not squares in Fq, and F4(q) 6 Ω−(26, q) otherwise (again assuming that β is non-

degenerate). Here, Jβ is the matrix of β with respect to some choice of basis for A,

as in Definition 2.5.10.

Let t ∈ 0, 1, . . . , 6, and recall that the basis vectors for O satisfy it = −it,i2t = −1 and 1 = 1. The following 26 trace 0 matrices are linearly independent


elements of A, and so they define a basis for A.

e1 :=

1 0 0

0 0 0

0 0 −1

, e2 :=

0 0 0

0 1 0

0 0 −1

, e3 :=

0 0 1

0 0 0

1 0 0

,

e4+t :=

0 0 −it0 0 0

it 0 0

, e11 :=

0 1 0

1 0 0

0 0 0

, e12+t :=

0 it 0

−it 0 0

0 0 0

,

e19 :=

0 0 0

0 0 1

0 1 0

, and e20+t :=

0 0 0

0 0 it

0 −it 0

.

We have (ej, ej)β = 2 for all j ∈ 1, . . . , 26, (e1, e2)β = (e2, e1)β = 1, and

(ej, ek)β = 0 for all other values of j, k ∈ 1, . . . , 26. In particular, for distinct

integers s, t ∈ 0, 1, . . . , 6,

e4+t e4+s =

−12(itis + isit) 0 0

0 0 0

0 0 −12(itis + isit)

= 0,

since isit = −itis. Similarly, e12+t e12+s = 0 = e20+t e20+s. Since β is nonzero, and

since A is an irreducible F4(q)-module, β is indeed non-degenerate by Proposition

2.8.3. Furthermore, with respect to the given basis, Jβ is the 26-dimensional matrix

2 1

1 2

2. . .

2

,

where non-diagonal entries that have been left blank are equal to 0. Thus

det(Jβ) = 2 det

2

2. . .

2

− det

1

2. . .

2

,

where the matrices on the right hand side of this equation are 25-dimensional. We

therefore have det(Jβ) = 226− 224 = 224 · 3 = (212)2 · 3. Hence det(Jβ) is a square in

Fq if and only if 3 is a square in Fq. Using Proposition 5.4.3, we see that F4(q) is a

subgroup of Ω+(26, q) if and only if q ≡ 1, 7 (mod 12), and that F4(q) is a subgroup

of Ω−(26, q) if and only if q ≡ 5, 11 (mod 12).

When (G, d) = (F4(q), 26) with p > 3, A is an irreducible 26-dimensional

Fq[F4(q)]-module by Theorem 5.4.6. Thus Lemma 4.6.8 implies that G is conju-

gate in GL(26, q) to the group F4(q) from Theorem 5.4.6. We can also identify the


G-module V with the F4(q)-module A, up to twisting by a field automorphism,

with V ∼= A as vector spaces. Since F4(q) preserves a non-degenerate orthogonal

form on A, Proposition 2.5.13 implies that G also preserves a non-degenerate or-

thogonal form of the same type on V . Similarly, when p = 3, we can identify the

G-module V with the F4(q)-module A/〈I〉, up to twisting by a field automorphism,

with V ∼= A/〈I〉 as vector spaces.

The following proposition generalises part of the above discussion to the general

case of (G, d) ∈ (F4(q), c), (E8(q), 248), with p > 3.

Proposition 5.4.8 ([53, Table 5.4.C]). The group G preserves a non-degenerate

orthogonal form on V .

Throughout the remainder of this section, let β be the non-degenerate orthogonal

form on V preserved by G, and let ε be the type of β. If G = F4(q) with p > 3,

then ε ∈ +,− is given in Lemma 5.4.7. Similarly, if G = E8(q), then d = 248 is

even, and hence ε ∈ +,−. However, we will not determine the type of β in this

case. On the other hand, if G = F4(q) with p = 3, then d = 25 is odd and hence

ε = . In each case, G lies in the general orthogonal group GOε(d, q) that preserves

β. Moreover (as we have seen in the case of G = F4(q) with p > 3), the perfect

group G lies in (GOε(d, q))∞, which is equal to Ωε(d, q) by Proposition 2.6.10.

Lemma 5.4.9. The normaliser N of G in GL(d, q) is ZGLG. Moreover, N is not

maximal in GL(d, q).

Proof. The cosets of Inn(G) in Aut(G) are defined by the field automorphisms of G

[73, p. 161, p. 177]. Lemma 4.6.8 shows that if α is a nontrivial field automorphism

of G, then the Fq[G]-module V is not isomorphic to V α. It follows from Lemma

5.1.7 that N = ZGLG. Finally, G is a subgroup of Ωε(d, q) by Lemma 5.4.7. By

considering group orders [13, Ch. 1.6.4], we have N < ZGLΩε(d, q) < GL(d, q), and

therefore N is not maximal in GL(d, q).

We now determine the maximal subgroups of SL(d, q) and Ωε(d, q) that contain

G. Theorem 5.1.6 shows that, as part of this, we must consider C9-subgroups H such

that H∞/Z(H∞) is not isomorphic to a simple group of Lie type defined over a field

of characteristic p. Note that H∞ is absolutely irreducible by Definition 2.10.3 and

quasisimple by Lemma 2.10.6. The following three lemmas, which we derive from

the tables in [38], list the possible groups H∞ (with p still assumed to be odd).

Lemma 5.4.10. Let F be a field of characteristic p, and let X be a finite, quasisim-

ple, absolutely irreducible subgroup of GL(25,F). Moreover, suppose that X/Z(X)

is not isomorphic to a simple group of Lie type defined over a field of characteristic

p. Then X ∈ A26, A27,PSL(2, 25),PSL(2, 49),PSp(4, 7), 3D4(2).


Lemma 5.4.11. Suppose that p > 3, let F be a field of characteristic p, and let

X be a finite, quasisimple, absolutely irreducible subgroup of GL(26,F). Moreover,

suppose that X/Z(X) is not isomorphic to a simple group of Lie type defined over

a field of characteristic p. Then X is a group in the set

A27, A28,PSL(2, 25), 2·PSL(2, 25),PSL(2, 27), 2·PSL(2, 27), 2·PSL(2, 53),PSL(3, 3),

2·PSL(3, 4),PSL(4, 3),PSU(3, 3),PSp(6, 2), 3D4(2), 2F4(2)′.

Lemma 5.4.12. Let F be a field of characteristic p, and let X be a finite, quasisim-

ple, absolutely irreducible subgroup of GL(248,F). Moreover, suppose that X/Z(X)

is not isomorphic to a simple group of Lie type defined over a field of characteristic

p. Then X ∈ A249, A250, 2·PSL(4, 5),Th.

A given group in each of these sets may only be an absolutely irreducible sub-

group of GL(d,F) for particular values of p. Note that |2F4(2)′| = 17971200 [53,

Ch. 5.1], and that the non-abelian simple Thompson group Th has order 215 · 310 ·53 · 72 · 13 · 19 · 31 [73, Ch. 5.8.7].

Theorem 5.4.13. The only maximal subgroup of SL(d, q) that contains G is the

full group of similarities of β in SL(d, q). This group of similarities is isomorphic

to SOε(d, q).[(q − 1, d)]. Additionally, the only maximal subgroup of Ωε(d, q) that

contains G is the C9-subgroup NΩε(d,q)(G) = Z(Ωε(d, q))G.

Proof. As G preserves the non-degenerate orthogonal form β, Lemma 5.1.5 implies

that the only maximal geometric subgroup of SL(d, q) that contains G is the full

group of similarities of β in SL(d, q). Furthermore, this group is isomorphic to

SOε(d, q).[(q − 1, d)] [13, Table 2.11]. We also have from Lemma 5.1.5 that no

geometric maximal subgroup of Ωε(d, q) contains G. By Theorem 2.10.5, any other

maximal subgroup of S ∈ SL(d, q),Ωε(d, q) that contains G is a maximal C9-

subgroup H. Definition 2.10.3 and Lemma 2.10.6 imply that H∞ is absolutely

irreducible and quasisimple. Thus if H∞/Z(H∞) is not isomorphic to a simple

group of Lie type defined over a field of characteristic p, then H∞ is listed in Lemma

5.4.10, 5.4.11 or 5.4.12. Using the tables in Section 4.1, we see that, for each q, |G|does not divide the order of any of these listed groups. Hence Lagrange’s Theorem

implies that G 66 H∞, and so the perfect group G does not lie in H. Thus we

require H = NS(G) by Theorem 5.1.6. Furthermore, since G preserves β, this

theorem shows that no maximal C9-subgroup of SL(d, q) contains G. By considering

group orders [13, Ch. 1.6.4], we see that G < Ωε(d, q), and so NΩε(d,q)(G) must be

a maximal subgroup of Ωε(d, q) containing G. Finally, Lemma 5.4.9, Proposition

2.6.9 and Dedekind’s Identity give N εΩ(d, q)(G) = Z(Ωε(d, q))G.

If we set H = G and S = Ωε(d, q), then the above results show that the hy-

potheses of Proposition 5.1.8 are satisfied, with B = 1 and M equal to the maximal

5.5. Overgroups of E6(q) or 3·E6(q) in GL(27, q) 119

GL(d, q)

Z(GL(d, q))Ωε(d, q)

Z(GL(d, q))GNGL(d,q)(G) =

Z(SL(d, q))GNSL(d,q)(G) =

SL(d, q)

SOε(d, q).[(q − 1, d)]

SOε(d, q)

Ωε(d, q)

Z(Ωε(d, q))G

G

Figure 5.4.1: Some overgroups of G in GL(d, q), where q is a power of an odd primep, and where (G, d) ∈ (F4(q), c), (E8(q), 248), with c = 25 if p = 3 and c = 26otherwise. Double edges indicate maximal containment, and two given subgroupsconnected by a dashed edge may be equal for certain values of d and q.

subgroup of SL(d, q) that preserves β up to scalars. Proposition 5.1.8 therefore

implies the following.

Proposition 5.4.14. The subgroups of SL(d, q) that contain G but do not con-

tain Ωε(d, q) are the subgroups K such that G 6 K 6 NSL(d,q)(G). Additionally,

NSL(d,q)(G) = ZSLG.


GL(d, q). Recall from Definition 2.6.7 that Ωε(d, q) is maximal in SOε(d, q). Addi-

tionally, Proposition 2.6.9 shows that if Ωε(d, q) is not simple, then |Z(Ωε(d, q))| = 2,

in which case G is maximal in Z(Ωε(d, q))G. Otherwise, Z(Ωε(d, q)) is trivial, in

which case G = NΩε(d,q)(G) is maximal in Ωε(d, q). In particular, F4(q) is maximal

in Ω(25, q) when p = 3.

5.5 Overgroups of E6(q) or 3·E6(q) in GL(27, q)

In this section, we explore the overgroups of G in GL(d, q), where G = E6(q)

and d = 27. Corollary 4.2.3 implies that G ∼= G when q 6≡ 1 (mod 3). For general

q, Table 4.1.3 shows that, for the universal cover J of G, |Z(J)| = (3, q − 1). Since

the Sylow p-subgroup of Z(J) is trivial, Lemma 4.2.1 gives G = J . In addition,

G ∼= J/Z(J) = G/Z(G) by Theorem 2.2.4. Hence |G| = 3|G| when q ≡ 1 (mod 3).

Therefore, for all q, Table 4.1.3 gives |G| = q36∏

i∈2,5,6,8,9,12(qi − 1). In fact, when

q ≡ 1 (mod 3), we have G = 3·E6(q) [73, Ch. 4.10.6]. Furthermore, since G is

perfect, it is a subgroup of (GL(27, q))′, which is SL(27, q) by Proposition 2.6.2. We

also have |G| < |SL(27, q)| [13, Ch. 1.6.4], and thus G < SL(27, q).

Proposition 5.5.1. The group G does not preserve a σ-Hermitian or reflexive bi-

linear form on V .


Proof. Since V is an irreducible G-module, we have from Proposition 2.8.3 that

any nonzero σ-Hermitian or reflexive bilinear form on V preserved by G is non-

degenerate. Additionally, Lemma 5.1.1 implies that such a non-degenerate form is

only preserved by G if there exists an isomorphism from the dual G-module V ∗ to V ,

or from V ∗ to V φ for some nontrivial field automorphism φ of G. However, Lemma

4.6.8 shows that no such isomorphism exists, and hence the result follows.

Proposition 5.5.2. The group G is a C9-subgroup of SL(27, q).

Proof. Recall from the discussion at the start of Section 5.1 that no conjugate of

the absolutely irreducible, perfect group G can be defined over a proper subfield

of Fq. In addition, Proposition 5.1.2 implies that G/(G ∩ ZGL) = G/Z(G) ∼= G is

simple, and G does not preserve a σ-Hermitian or reflexive bilinear form on F27q by

Proposition 5.5.1. Therefore, the proper subgroup G of SL(27, q) lies in class C9 by

Definition 2.10.3.

Lemma 5.5.3. The normaliser N of G in GL(27, q) is (ZGLG).(q− 1, 3). Moreover,

N is a (proper) C9-subgroup of GL(27, q).

Proof. First, as G is simple, Lemma 2.2.6 implies that the automorphisms of G lift

to the automorphisms of the universal cover of G, via a 1-1 correspondence. From

the discussion at the start of this section, this universal cover is G. It follows from

Wilson’s [73, p. 172] classification of the outer automorphisms of G that the cosets

of Inn(G) in Aut(G) are defined by the field and graph automorphisms of G, and

when q ≡ 1 (mod 3), also by the two nontrivial diagonal automorphisms of G of

order 3 (with each equal to the inverse of the other). Lemma 4.6.8 shows that if α

is a nontrivial field or graph automorphism of G, then the Fq[G]-module V is not

isomorphic to V α. However, if α is a diagonal automorphism of G, then V ∼= V α

[13, Proposition 5.1.9]. It follows from Lemma 5.1.7 that |N/(ZGLG)| = (q − 1, 3),

and hence N = (ZGLG).(q − 1, 3).

Now, since N/(ZGLG) is soluble, N∞ = (ZGLG)∞, which is equal to (G)∞.

As G is a C9-subgroup of GL(27, q) by Proposition 5.5.2, N∞ satisfies all prop-

erties required by Definition 2.10.3. Additionally, by considering group orders

[13, Ch. 1.6.4], we see that N does not contain SL(27, q). This also implies that

N < GL(27, q). Finally, it follows from the Third Isomorphism Theorem that

(N/ZGL)/(ZGLG/ZGL) ∼= N/(ZGLG), which is isomorphic to the group of order

(q − 1, 3). Hence N/ZGL = (ZGLG/ZGL).(q − 1, 3). The Second Isomorphism Theo-

rem gives ZGLG/ZGL∼= G/(G∩ZGL), which is equal to G/Z(G) by Proposition 5.1.2,

and which is in turn isomorphic to G by Lemma 4.2.1. Thus N/ZGL∼= G.(q− 1, 3).

The group G.(q−1, 3) is isomorphic to the subgroup of Aut(G) generated by Inn(G)

and the diagonal automorphisms of G (i.e., the automorphisms that lift to the diag-

onal automorphisms of G). Hence N/ZGL is almost simple by Definition 2.2.1, and

so N is a C9-subgroup of GL(27, q) by Definition 2.10.3.

5.6. Overgroups of 2·E7(q) in GL(56, q) 121

GL(27, q)

(Z(GL(27, q))G).(q − 1, 3)NGL(27,q)(G) = SL(27, q)

NSL(27,q)(G)

G

Figure 5.5.1: Some overgroups of G in GL(27, q), for an odd prime power q, whereG = 3·E6(q) if q ≡ 1 (mod 3) and G = E6(q) otherwise. Double edges indicatemaximal containment, and subgroups connected by dashed edges are equal whenq 6≡ 1 (mod 3).

In the proof of the following theorem, J1 denotes the non-abelian simple Janko

group of order 175560 [73, Ch. 5.9.1]. Recall also that |2F4(2)′| = 17971200.

Theorem 5.5.4. The only maximal subgroup of SL(27, q) that contains G is the

C9-subgroup NSL(27,q)(G).

Proof. As G is a proper subgroup of SL(27, q), there exists a maximal subgroup

of SL(27, q) that contains G. Let H be such a maximal subgroup. Since G does

not preserve a nonzero σ-Hermitian or reflexive bilinear form on V , we have from

Lemma 5.1.5 that H is not a geometric subgroup of SL(27, q). Hence H is a C9-

subgroup of SL(27, q) by Theorem 2.10.5. Furthermore, H∞ is absolutely irreducible

and quasisimple by Definition 2.10.3 and Lemma 2.10.6.

Using the tables in [38], we see that if H∞/Z(H∞) is not isomorphic to a simple

group of Lie type defined over a field of characteristic p, then H∞ lies in the set

A9, A28, A29,PSL(2, 27),PSL(2, 53),PSL(3, 3),

PSU(3, 3),PSp(6, 2), 3·Ω(7, 3), 3·G2(3), 2F4(2)′, J1.

The tables in Section 4.1 show that, for all q, the order of each group in this set

is smaller than |G|. Hence Lagrange’s Theorem gives G 66 H∞, and so the perfect

group G does not lie in H. Thus we require H = NSL(27,q)(G) by Theorem 5.1.6.

We summarise our results about the overgroups of G in GL(27, q) in Figure

5.5.1, which is much simpler than the figures corresponding to the other excep-

tional Chevalley groups. It is clear from Lemma 5.5.3 and Dedekind’s Identity that

NSL(27,q)(G) = ZSLG when q 6≡ 1 (mod 3), and thatNSL(27,q)(G) ∈ ZSLG, (ZSLG).3when q ≡ 1 (mod 3). In the former case, we have ZSL = 1, and hence G ∼= E6(q) is

a maximal subgroup of SL(d, q).

5.6 Overgroups of 2·E7(q) in GL(56, q)

We now consider the overgroups of G in GL(d, q), where G = E7(q) and d =

56. Table 4.1.3 shows that, for the universal cover J of G, |Z(J)| = 2. Since


the Sylow p-subgroup of Z(J) is trivial, G = J by Lemma 4.2.1. We also have

G ∼= J/Z(J) = G/Z(G) from Theorem 2.2.4. Hence |G| = 2|G|, and so Table 4.1.3

gives |G| = q63∏

i∈2,6,8,10,12,14,18(qi − 1). In fact, G = 2·E7(q) [73, Ch. 4.12]. As G

is perfect, it is a subgroup of (GL(56, q))′, which is SL(56, q) by Proposition 2.6.2.

Proposition 5.6.1 ([53, Proposition 5.4.18]). The group G is a subgroup of the

symplectic group Sp(56, q).

Thus, by Definition 2.6.3, G preserves a non-degenerate alternating form β on

V . As in [13, Definition 1.6.14], we denote the largest group of similarities of β in

GL(56, q) by CSp(56, q).

Lemma 5.6.2. The normaliser N of G in GL(56, q) is (ZGLG).2. Moreover, N is a

proper subgroup of CSp(56, q).

Proof. As G is simple, Lemma 2.2.6 implies that the automorphisms of G lift to

the automorphisms of the universal cover of G, via a 1-1 correspondence. From

the discussion at the start of this section, this universal cover is G. It follows from

Wilson’s [73, p. 177] classification of the outer automorphisms of G that the cosets

of Inn(G) in Aut(G) are defined by the field automorphisms of G and the diagonal

automorphism of G of order 2. Lemma 4.6.8 shows that if α is a nontrivial field

automorphism of G, then the Fq[G]-module V is not isomorphic to V α. However,

if α is the diagonal automorphism of G, then V ∼= V α [13, Proposition 5.1.9]. It

follows from Lemma 5.1.7 that |N/(ZGLG)| = 2, and hence N = (ZGLG).2. Finally,

since the absolutely irreducible group G preserves the non-degenerate alternating

form β, Lemma 2.9.14 implies that N 6 CSp(56, q). By considering group orders

[13, Ch. 1.6.4], we see that N < CSp(56, q).

In the proof of the following theorem, M22 denotes the non-abelian simple Math-

ieu group of order 443520; HS denotes the non-abelian simple Higman-Sims group

of order 44352000; and J2 denotes the non-abelian simple Hall-Janko group of order

604800 [73, Ch. 5.2.9, Ch. 5.5.1, Ch. 5.6.4]. Recall also that |J1| = 175560.

Theorem 5.6.3. The only maximal subgroup of SL(56, q) that contains G is the

full group of similarities of β in SL(56, q). This group of similarities is isomorphic to

[(q − 1, 56)]·PSp(56, q) when 8 | (q − 1), and to [(q − 1, 56)]·PCSp(56, q) otherwise.

Additionally, the only maximal subgroup of Sp(56, q) that contains G is the C9-

subgroup NSp(56,q)(G).

Proof. As G preserves the non-degenerate alternating form β, Lemma 5.1.5 implies

that the only maximal geometric subgroup of SL(56, q) that contains G is the full

group of similarities of β in SL(56, q). Bray, Holt and Roney-Dougal [13, Table 2.11]

show that the isomorphism type of this maximal subgroup is as required. We also

have from Lemma 5.1.5 that no geometric maximal subgroup of Sp(56, q) contains G.

5.6. Overgroups of 2·E7(q) in GL(56, q) 123

By Theorem 2.10.5, any other maximal subgroup of S ∈ SL(56, q), Sp(56, q) that

contains G is a maximal C9-subgroup H. In particular, H∞ is absolutely irreducible

and quasisimple by Definition 2.10.3 and Lemma 2.10.6.

Using the tables in [38], we see that if H∞/Z(H∞) is not isomorphic to a simple

group of Lie type defined over a field of characteristic p, then H∞ is in the set

A8, 2·A8, A9, 2·A9, A10, 2·A10, 2·A11, A57, A58, 2·PSL(2, 113), 4·PSL(3, 4),PSL(3, 7),

PSU(3, 8), 2·PSU(4, 3), 2·PSU(6, 2),PSp(6, 2), 2·PSp(6, 2), 2·PΩ+(8, 2), 2·(2B2(8)),

2·M22, 4·M22, 2·HS, J1, 2·J2.

The tables in Section 4.1 show that, for each q, |G| does not divide the order of any

group in this set. Thus Lagrange’s Theorem implies that G 66 H∞, and hence the

perfect group G does not lie in H. We therefore require H = NS(G) by Theorem

5.1.6. Furthermore, since G preserves β, this theorem shows that no maximal C9-

subgroup of SL(56, q) contains G. By considering group orders [13, Ch. 1.6.4], we see

that G < Sp(56, q), and hence NSp(56,q)(G) must be a maximal subgroup of Sp(56, q)

containing G.

If we set H = G and S = Sp(56, q), then the above results show that the

hypotheses of Proposition 5.1.8 are satisfied, with B equal to the group of order 2

and M equal to the maximal subgroup of SL(56, q) that preserves β up to scalars.

Proposition 5.1.8 therefore implies the following.

Proposition 5.6.4. The subgroups of SL(56, q) that contain G but do not contain

Sp(56, q) are the subgroups K such that G 6 K 6 NSL(56,q)(G).


GL(56, q). Note that Lemma 5.6.2, Proposition 2.6.4 and Dedekind’s Identity im-

ply that NSp(56,q)(G) ∈ Z(Sp(56, q))G, (Z(Sp(56, q))G).2, and that NSL(56,q)(G) ∈ZSLG, (ZSLG).2.

GL(56, q)

CSp(56, q)

(Z(GL(56, q))G).2NGL(56,q)(G) =

NSL(56,q)(G)

SL(56, q)

CSp(56, q) ∩ SL(56, q)

Sp(56, q)

NSp(56,q)(G)

G

Figure 5.6.1: Some overgroups of G = 2·E7(q) in GL(56, q), for an odd prime powerq. Double edges indicate maximal containment, and two given subgroups connectedby a dashed edge may be equal for certain values of q.

Chapter 6

Proof of the main theorem

6.1 Stabilisers of subspaces of Lie powers

In this chapter, we will use the results of the previous chapters to precisely

state and prove the main theorem of this thesis. We retain the notation outlined in

Hypothesis 5.1.1. Before stating our main theorem, we will prove several necessary

results about the stabilisers in GL(d, q) of particular subspaces of L2V , L3V and

V ⊕ L2V . Recall from Lemma 2.11.8 that there is an Fq[GL(d, q)]-isomorphism

between L2V and A2V (as p is odd), and between L3V and (A2V ⊗ V )/A3V when

p > 3. We will use this fact throughout this chapter without further reference.

Additionally, all results in this chapter are our own.

Recall from Proposition 5.6.1 that if G = E7(q), then G is a subgroup of the

symplectic group Sp(56, q). As in Section 5.6, we write CSp(56, q) to denote the

largest group of similarities in GL(56, q) of the non-degenerate alternating form

preserved by Sp(56, q). In Section 4.5, we claimed that applying Theorem 3.2.10(ii)

to the simply connected version of E7(p) is more useful than applying Theorem

3.2.10(i). The following lemma explains the reason for this.

Lemma 6.1.1. Suppose that G = E7(q).

(i) The groups G and CSp(56, q) stabilise the same set of subspaces of L2V .

(ii) Suppose that p > 3. Then each of Sp(56, q) and CSp(56, q) stabilises exactly

two nontrivial proper subspaces of L3V , of dimension 56 and 58464, respec-

tively. If p = 19, then the latter subspace contains the former, and otherwise,

L3V splits as the direct sum of these subspaces.

Proof. Let X be the simple group of Lie type PSp(56, q), which is denoted C28(q)

in Lie notation. The linear algebraic group related to X is therefore X = C28, and

the simply connected version X of X defined in Lemma 4.2.1 is Sp(56, q) [60, p. 193].

Let K be the algebraic closure of the field Fp. Up to isomorphism, twisting by a

field automorphism, and duality, the module with highest weight1 λ1, of dimension

56, is the unique minimal K[X]-module [55, §1], which we will denote by U . In fact,

1The weight associated with C28 that we denote λi is denoted λ29−i by Lubeck [56].

125

126 Chapter 6. Proof of the main theorem

this module is self-dual [60, p. 132–133], and hence it is the unique minimal K[X]-

module up to isomorphism and twisting by a field automorphism. It follows from

Theorem 4.6.4 that there is a unique (in the same way) minimal Fq[X]-module, of

dimension 56. As G is a subgroup of X, it follows that the irreducible 56-dimensional

G-module V is the restriction to G of a minimal Fq[X]-module W . Additionally, for

each r ∈ 2, 3, we see from the definition of the action of X on LrW in Proposition

2.11.7 that LrV is the restriction to G of LrW .

Suppose now that p /∈ 2, 7. Then L2U has two composition factors, of dimen-

sion 1 and 1539, respectively [55, §1]. Hence L2U is multiplicity free by Theorem

4.3.5, and thus L2W is also multiplicity free by Theorem 4.6.4. In particular, L2W

is the direct sum of a 1-dimensional submodule and a 1539-dimensional submodule.

By Theorem 4.6.11, G and Sp(56, q) stabilise the same set of subspaces of L2V .

Moreover, Sp(56, q) C6 CSp(56, q) [53, p. 14]. It follows from Corollary 2.9.5 that

CSp(56, q) and Sp(56, q) stabilise the same set of subspaces of L2V .

If instead p = 7, then the G-module L2V is uniserial, and it has three composition

factors, of dimension 1, 1 and 1538, respectively, by Theorem 4.6.11. In fact, these

are exactly the dimensions of the Sp(56, q)-composition factors of L2W [55, §1].

Hence Sp(56, q) is uniserial, and in particular, G and Sp(56, q) stabilise the same

set of subspaces of L2V . Since CSp(56, q) normalises Sp(56, q), and since Sp(56, q)

stabilises no two equidimensional subspaces of L2V , each CSp(56, q)-composition

series for L2V restricts to a Sp(56, q)-composition series, by Corollary 2.9.7. As L2V

has a unique Sp(56, q)-composition series, it follows that CSp(56, q) and Sp(56, q)

stabilise the same set of subspaces of L2V .

We now apply the methods of Section 4.5 and the Magma code in Appendix A.3

to the K[X]-modules U and L3U . In particular, calculations using this code show

that the Weyl orbit of λ1 has size 56. Since the weight multiset for U must contain

exactly 56 weights by Corollary 4.2.27, and since it must contain the Weyl orbit of

λ1 by Lemma 4.2.24, it follows that the weight multiset of U is precisely this Weyl

orbit. We also observe using Magma that the highest weight of L3U is λ1 +λ2. The

computations described in [56, §3] can be used to show that if p /∈ 3, 19, then

the irreducible module L(λ1 + λ2) has dimension 58464, and the weight multiset of

this module consists of one, two and 54 copies of the Weyl orbits of the weights

λ1 + λ2, λ3 and λ1, respectively [57]. When these weights are excluded from the

weight multiset of L3U , 56 weights remain, the highest of which is λ1. It follows

that when p /∈ 3, 19, L3U has two composition factors, of dimension 56 and 58464,

respectively. In this case, L3U is multiplicity free by Theorem 4.3.5, as is L3W by

Theorem 4.6.4. Specifically, L3W is the direct sum of a submodule of dimension

56 and a submodule of dimension 58464, and these are the only nonzero proper

submodules of L3W by Corollary 2.8.16. As above, Corollary 2.9.5 implies that

CSp(56, q) stabilises the same subspaces of L3V as Sp(56, q).

6.1. Stabilisers of subspaces of Lie powers 127

Finally, when p = 19, the irreducible module L(λ1 + λ2) has dimension 58408,

and the weight multiset of this module consists of one, two and 53 copies of the Weyl

orbits of the weights λ1 + λ2, λ3 and λ1, respectively [57]. When these weights are

excluded from the weight multiset of L3U , 112 weights remain, the highest of which

is λ1, with multiplicity 2. Hence L3W has two 56-dimensional composition factors

and one 58408-dimensional composition factor. The G-submodule structure of L3V

given in Figure 4.6.6 for the case of q = 19 implies that L3W is uniserial, with the

dimensions of its submodules as required. A very similar argument to the one used

in the uniserial cases of Theorem 4.6.11(i) shows that the submodule structure of

L3W is as required even when q > p = 19. As above, Corollary 2.9.7 implies that

CSp(56, q) and Sp(56, q) stabilise the same set of subspaces of L3V .

We are therefore not able to distinguish between the simply connected version of

E7(q) and CSp(56, q) by considering how these groups act on L2V . Thus applying

Theorem 3.2.10(i) to these two groups (with q = p) yields the same p-group.

Proposition 6.1.2. Let r ∈ 2, 3, and suppose that p > r. Then G and ZGLG

stabilise the same set of subspaces of LrV . Furthermore, if there is a subspace of

LrV that is stabilised by G but not by NGL(d,q)(G), then r = 3, and:

(i) G = E6(q), p = 5, and q ≡ 1 (mod 3); or

(ii) G = E7(q), p ∈ 7, 11, 19, and q > p.

Proof. Let z ∈ ZGL, so that z = µId, where µ ∈ Fq \ 0 and where Id is the d× didentity matrix over Fq. Then Proposition 2.11.7 implies that for each v ∈ LrV , we

have vz = µrv. Hence ZGL stabilises every subspace of LrV , and so ZGLG stabilises

exactly the set of subspaces of LrV that are stabilised by G.

Next, let N := NGL(d,q)(G). If G ∈ G2(q), F4(q), E8(q), or if G = E6(q) with

q 6≡ 1 (mod 3), then Lemmas 5.3.2, 5.4.9 and 5.5.3 imply that N = ZGLG, and so N

and G stabilise the same set of subspaces of LrV . Otherwise, if LrV is multiplicity

free, then Theorem 4.6.11 shows that no two irreducible submodules of LrV are

equidimensional. We also have ZGLG C6 N , as N normalises ZGL and G. Thus

Corollary 2.9.5 implies that N and ZGLG stabilise the same set of subspaces of LrV .

We now consider the remaining cases where LrV is not multiplicity free, and

either (i) and (ii) do not apply, or r = 2. Theorem 4.6.11 implies that G = E7(q),

with p = 7 if r = 2, and with q = p ∈ 7, 11, 19 if r = 3. Recall from Lemma 5.6.2

that N < CSp(56, q). Hence Lemma 6.1.1 implies that N and G stabilise the same

set of subspaces of L2V , as required, and that N stabilises the subspaces of L3V of

dimension 56 and 58464. Figures 4.6.4–4.6.6 show that no two submodules of L3V

are equidimensional, and so we will write Uk to denote the unique submodule of

L3V of dimension k, when such a submodule exists. Furthermore, ZGLG is a normal

subgroup of N of index 2 by Lemma 5.6.2, and this index is coprime to p. In order to


show that N stabilises all submodules of L3V , we will use the submodule structure of

L3V shown in Figures 4.6.4–4.6.6; the fact that this is exactly the ZGLG-submodule

structure of L3V ; the fact that N stabilises L3V , U56 and U58464; and the following:

(a) if N stabilises G-submodules X and U of L3V , and if there exists a G-

submodule W of L3V such that X = U ⊕W , then N stabilises W ;

(b) if N stabilises G-submodules U and W of L3V , with W ⊆ U , then N stabilises

each G-submodule in some G-composition series for U containing W ; and

(c) if N stabilises a multiplicity free G-submodule U of L3V , then N stabilises

each G-submodule of U .

Properties (a) and (c) are immediate consequences of Corollaries 2.9.3 and 2.9.5,

respectively, while (b) follows from Corollary 2.9.7 and the fact that each submodule

of an N -module lies in some composition series for the N -module.

Suppose first that p = 7. We see from Figure 4.6.4 that each G-composition

series for L3V that contains U56 also contains either U52040, U7448, or both U6536 and

U57608. By (b), N stabilises at least one of these composition series. As U52040 =

U56⊕U912⊕U51072 is multiplicity free, (c) implies that if N stabilises U52040, then it

also stabilises U51072. In addition, L3V = U7448 ⊕U51072 and U57608 = U6536 ⊕U51072,

and hence if N stabilises either U7448 or both U6536 and U57608, then it also stabilises

U51072 by (a). Therefore, N does indeed stabilise U51072. It follows that N stabilises

U51128 = U56 ⊕ U51072. As L3V = U51128 ⊕ U7392, N also stabilises U7392 by (a). The

submodule U7392 is uniserial, and hence it has a unique G-composition series, which

(b) implies is stabilised by N . Finally, L3V = U56⊕U7392⊕U51072, and N stabilises

each G-submodule of each direct summand. Therefore, N stabilises all direct sums

of G-submodules of these summands, which accounts for all G-submodules of L3V .

The argument in the case p = 11 is equivalent, but with each submodule from

Figure 4.6.4 replaced by its corresponding submodule from Figure 4.6.5, for example

with U51128 replaced by U968. The case where p = 19 is similar, but here we must

show that N stabilises the irreducible submodules U912 and U51072. For a contradic-

tion, suppose that N does not stabilise both of these irreducible modules. Then by

(c), N does not stabilise the multiplicity free submodule U52040 = U56⊕U912⊕U51072.

Figure 4.6.6 shows that each G-composition series for L3V that contains U58464, but

does not contain U52040, also contains either U7392 or U57552, and N stabilises at least

one of these composition series by (b). Since U7392⊕U51072 = U58464 = U57552⊕U912,

(a) implies that N stabilises at least one of U912 and U51072. If N stabilises U912,

then (b) implies that it stabilises a G-composition series for L3V that contains

U912, but does not contain U52040, and hence N stabilises U7392. However, as

U58464 = U7392 ⊕ U51072, N stabilises U51072 by (a). Similarly, if N stabilises U51072,

then (b) shows that N stabilises U57552, and it follows from (a) that N stabilises U912.

We have a contradiction in either case, and hence N does in fact stabilise both U912

6.1. Stabilisers of subspaces of Lie powers 129

and U51072. An equivalent argument to the p = 7 case then shows that N stabilises

each G-submodule of the uniserial submodule U6536, and hence N stabilises each

G-submodule of L3V = U912 ⊕ U6536 ⊕ U51072.

In particular, G and NGL(d,p)(G) stabilise the same set of subspaces of LrV

whenever q = p. In fact, if Conjecture 4.6.12 holds, then the arguments in the above

proof with G = E7(q) and p ∈ 7, 11, 19 hold even when q 6= p. Furthermore, if this

conjecture holds, then a similar argument using Figure 4.6.2 shows that NGL(d,p)(G)

stabilises all submodules of L3V whenever G = E6(q) and p = 5.

Lemma 6.1.3. Suppose that p > r, where r = 3 if G ∈ E6(q), E7(q), and r = 2

otherwise. Suppose also that q = p if G = E6(q) with p = 5, or if G = E7(q) with

p ∈ 7, 11, 19. Additionally, let X be a nonzero proper submodule of LrV . Then

either GL(d, q)X = NGL(d,q)(G), or G = E7(q) and CSp(56, q) stabilises X.

Proof. Suppose that CSp(56, q) does not stabilise X if G = E7(q). Let N :=

NGL(d,q)(G), and let M := NSL(d,q)(G). We showed in Chapter 5 that the normal sub-

group ZSLG of M has index at most 3. Thus M∞ = (ZSLG)∞ = G∞, which is equal

to G by Theorem 4.2.4. Hence G is a characteristic subgroup of M , which is normal

in NGL(d,q)(M). Thus NGL(d,q)(M) 6 N . Moreover, SL(d, q) is a normal subgroup

of GL(d, q) [73, p. 44], and so M = N ∩ SL(d, q) is a normal subgroup of N . This

implies that NGL(d,q)(M) = N . We also have that SL(d, q)X = GL(d, q)X∩SL(d, q) is

a normal subgroup of GL(d, q)X , and hence GL(d, q)X 6 NGL(d,q)(SL(d, q)X). Since

N stabilises X by Proposition 6.1.2, it suffices to show that SL(d, q)X = M . Note

that M 6 SL(d, q)X as M 6 N .

Suppose that G ∈ G2(q), F4(q), E8(q). Recall from Sections 5.3 and 5.4 that

G lies in SL(d, q) and preserves a non-degenerate orthogonal form on V . Let ε ∈,+,− be the type of this form. Then by Propositions 5.3.4 and 5.4.14, M is

the largest subgroup of SL(d, q) that contains G but does not contain Ωε(d, q). The

group Ωε(d, q) acts irreducibly on L2V [55, §1], and hence SL(d, q)X = M . Next,

suppose that G = E7(q). Then Proposition 5.6.4 implies that M is the largest

subgroup of SL(d, q) that contains G but does not contain Sp(56, q). Furthermore,

if Sp(56, q) stabilises X ⊂ L3V , then so does CSp(56, q) by Lemma 6.1.1. However,

we have assumed that this is not the case, and so SL(d, q)X = M .

Finally, suppose that G = E6(q). Then M is a maximal subgroup of SL(d, q) by

Theorem 5.5.4. We have from Lemma 2.11.8 that GL(d, q) acts irreducibly on L3V .

Clifford’s Theorem therefore implies that if the normal subgroup SL(d, q) of GL(d, q)

acts reducibly on L3V , then L3V is the direct sum of a set of proper equidimensional

subspaces that are stabilised by SL(d, q), and hence by G. However, Theorem 4.6.11

shows that no two G-submodules of L3V are equidimensional. Therefore, SL(d, q)

acts irreducibly on L3V , and hence SL(d, q)X = M .


If Conjecture 4.6.12 holds, then we do not need to assume that q = p in any case

in the above lemma.

Recall that if G ∈ (G2(q), E8(q), then G ∼= G by Corollary 4.2.3. The following

lemma will allow us to apply Theorem 3.3.9 to these groups (with q = p) to yield

p-groups that are not yielded by Theorem 3.2.10(i).

Lemma 6.1.4. Suppose that G ∈ (G2(q), E8(q). Then G ∼= G is the stabiliser in

GL(d, q) of a subspace of V ⊕ L2V that is isomorphic to L2V , and that does not

contain V or L2V . Moreover, no proper G-submodule of V ⊕ L2V has dimension

larger than dim(L2V ).

Proof. Theorem 4.6.11 shows that L2V contains a maximal submodule X such

that (L2V )/X ∼= V . Thus X is the kernel of a surjective Fq[G]-homomorphism from

L2V to V . We therefore have from Proposition 2.8.20 that V ⊕ L2V contains a

submodule M that is isomorphic to L2V , that does not contain V , and that cannot

be written as the direct sum of a submodule of V and a submodule of L2V . In

particular, L2V 6⊆M . Furthermore, the Second Isomorphism Theorem implies that

each composition factor of V ⊕ L2V is a composition factor of V or of L2V . It

follows from Theorem 4.6.11 that dim(L2V ) is the largest possible dimension of a

proper submodule of V ⊕ L2V .

Now, let H := GL(d, q)M . Since the subgroup G of H acts irreducibly on V , H

also acts irreducibly on V . Moreover, each H-submodule of L2V is a G-submodule

of L2V . Since H stabilises M , which cannot be written as the direct sum of a

submodule of V and a submodule of L2V , Theorem 2.8.19 implies that V is Fq[H]-

isomorphic to an H-composition factor of L2V . In particular, since dim(L2V ) >

dim(V ), H acts reducibly on L2V . Lemmas 5.3.2, 5.4.9 and 6.1.3 imply that G 6

H 6 ZGLG. We also require H∩ZGL = 1, since otherwise V is not Fq[H]-isomorphic

to an H-composition factor of L2V , by Proposition 2.8.24. Hence H = G. Note

that we do indeed have G ∩ ZGL = 1, as G is non-abelian and simple.

6.2 Inducing exceptional Chevalley groups on P/Φ(P )

In this section, we induce on the Frattini quotient of a p-group the simply con-

nected version G of a given exceptional Chevalley group G (defined over Fp), or the

normaliser of G in GL(d, p). As discussed at the start of Section 5.5 and the start

of Section 5.6, G = 3·G when G = E6(q) with q ≡ 1 (mod 3), and G = 2·G when

G = E7(q) (as p is odd). Otherwise, G ∼= G by Corollary 4.2.3.

Recall from Section 3.2 that if U is a proper subgroup of L2V , then PU is the

quotient of the universal p-group Γ(d, p, 2) by U , and if W is a proper subgroup of

L3V , then QW = Γ(d, p, 3)/W . Also recall from Section 3.1 that if P is a p-group of

rank d, then A(P ) denotes the subgroup of GL(d, p) induced by Aut(P ) on P/Φ(P ),

and that the exponent-p class of P must be at least 2 in order for A(P ) to be a proper

6.2. Inducing exceptional Chevalley groups on P/Φ(P ) 131

subgroup of GL(d, p). We begin by highlighting some proper subgroups of GL(d, p)

that cannot be induced on the Frattini quotient of a p-group of low exponent-p class

(in some cases, with low exponent or low nilpotency class also assumed).

Theorem 6.2.1. Suppose that G is defined over a field of odd prime order p, with

p > 3 if G ∈ E6(p), E7(p).

(i) Assume that

(G,H) ∈ (F4(p), ZGLF4(p)), (E6(p),GL(27, p)), (E7(p),CSp(56, p)),

and let K be a proper subgroup of H that contains G. Then there is no

p-group P of exponent-p class 2 such that A(P ) = K.

(ii) Assume that G ∈ G2(p), E8(p), and letK be a proper subgroup of ZGLG that

contains G. Then there is no p-group P of exponent-p class 2 and exponent p

such that A(P ) = K. Additionally, there is no abelian p-group P of exponent-p

class 2 such that A(P ) = K.

(iii) Assume that G ∈ E6(p), E7(p), and letK be a proper subgroup ofNGL(d,p)(G)

that contains G. Then there is no p-group P with A(P ) = K such that

P ∼= QW for some proper subspace W of L3V .

Proof.

(i) Observe from Theorem 4.6.11 that no composition factor of the G-module

L2V is isomorphic to the irreducible G-module V . Theorem 2.8.19 therefore

implies that the submodules of V ⊕ L2V are the submodules of L2V and the

direct sums of V and the submodules of L2V . It follows that any overgroup

of G in GL(d, p) that stabilises all G-submodules of L2V also stabilises all G-

submodules of V ⊕ L2V . If G = E6(p), then Theorem 4.6.11 shows that L2V

is irreducible, and thus each of its submodules is stabilised by H = GL(27, p).

In the other two cases, H stabilises all G-submodules of L2V by Lemma 6.1.1

and Proposition 6.1.2. We have shown that, in each case, there is no proper

subspace X of V ⊕ L2V such that K = GL(d, p)X . Hence Theorem 3.3.9

implies that there is no p-group P of exponent-p class 2 such that A(P ) = K.

(ii) Proposition 6.1.2 implies that ZGLG stabilises each G-submodule of L2V . This

means that there is no proper subspace U of L2V such that K = GL(d, p)U . It

follows from Theorem 3.3.9(ii) that there is no p-group P of exponent-p class

2 and exponent p such that A(P ) = K. Furthermore, V is an irreducible G-

module, and so its only proper subspace is 0, which is stabilised by GL(d, p).

Thus Theorem 3.3.9(i) implies that there is no abelian p-group P of exponent-p

class 2 such that A(P ) = K.


(iii) Proposition 6.1.2 implies that NGL(d,p)(G) stabilises each G-submodule of L3V .

Hence there is no proper subspace W of L3V such that K = GL(d, p)W .

Theorem 3.2.10(ii) therefore implies that there is no p-group P with A(P ) = K

such that P ∼= QW for some proper subspace W of L3V .

Although the first and second parts of Theorem 6.2.1 refer to the exponent-p

classes and exponents of p-groups, the third part of the theorem does not. Proposi-

tion 2.4.9 and Lemma 3.2.6 imply that if W is a proper subspace of L3V , then QW

has exponent-p class 3 and exponent p. However, there may exist a p-group P of

exponent-p class 3 and exponent p, with P not isomorphic to QW for any proper

subspace W of L3V , such that A(P ) is a group K defined in Theorem 6.2.1(iii).

Note that when G = E7(p), the subgroup K in Theorem 6.2.1(i) is a proper

subgroup of CSp(56, p). Indeed, Bamberg et al. [7, Table 6.1] constructed a p-group

P of exponent p and nilpotency class 2 (and hence exponent-p class 2) such that

A(P ) = CSp(n, p), for each integer n > 2 and each odd prime p such that CSp(n, p)

is a maximal subgroup of GL(n, p).

We are now able to state and prove the main theorem of this thesis, i.e., the full

version of Theorem 1.0.4. Here, we refer to optimal and quasi-optimal p-groups, as

defined in Definitions 3.1.1 and 3.2.11, respectively. Note that if P is an optimal or

quasi-optimal p-group with respect to a subgroup of GL(d, q), then P has rank d.

Theorem 6.2.2. Suppose that G is defined over a field of odd prime order p.

(i) If G ∈ G2(p), F4(p), E8(p), then each p-group that is optimal with respect

to NGL(d,p)(G) has exponent-p class 2, nilpotency class 2 and exponent p.

(ii) If G ∈ G2(p), E8(p), then each p-group that is optimal with respect to G

has exponent-p class 2, nilpotency class 2 and exponent p2.

(iii) If G ∈ E6(p), E7(p) and p > 3, then each p-group that is optimal (or quasi-

optimal) with respect to NGL(d,p)(G) has exponent-p class 3, nilpotency class

3 and exponent p.

Table 6.2.1 specifies the properties of each optimal p-group in case (i) or (ii), and

each quasi-optimal p-group in case (iii). Finally, each p-group in case (i) or (ii) has

a unique proper nontrivial characteristic subgroup.

Proof. Theorem 4.6.11 gives the dimensions of the proper G-submodules of L2V

(respectively, L3V ) in case (i) (respectively, in case (iii)). By Lemmas 6.1.1 and 6.1.3,

N := NGL(d,p)(G) is the stabiliser in GL(d, p) of the largest such proper submodule,

except when G = E7(p), in which case N is the stabiliser in GL(d, p) of the second

largest proper submodule. In case (ii), Lemma 6.1.4 implies that G is the stabiliser

in GL(d, p) of a proper subspace of V ⊕L2V that is isomorphic to L2V and does not

contain V or L2V , and that there is no proper G-submodule of V ⊕ L2V of larger

6.2. Inducing exceptional Chevalley groups on P/Φ(P ) 133

Table 6.2.1: The properties of each optimal or quasi-optimal p-group P from The-orem 6.2.2. Here, ZGL := Z(GL(d, p)), t := (3, p − 1), and r denotes both theexponent-p class and nilpotency class of P .

G d rExponent

of P|P | A(P )

G2(p) 7 2 p p14 ZGLG2(p)

G2(p) 7 2 p2 p14 G2(p)

F4(3) 25 2 3 377 ZGLF4(3)

F4(p), p > 3 26 2 p p78 ZGLF4(p)

E6(p), p > 3 27 3 p p456 (ZGL(t·E6(p))).t

E7(p), p > 3 56 3 p p2508 (ZGL(2·E7(p))).2

E8(p) 248 2 p p496 ZGLE8(p)

E8(p) 248 2 p2 p496 E8(p)

dimension. Let X be the specified proper submodule whose stabiliser in GL(d, p) is

N or G. By Theorem 3.2.10, N is equal to A(PX) or A(QX) in case (i) or case (iii),

respectively. In case (ii), Theorem 3.3.9 shows that G is equal to A(E∗/X), where

E∗ is the p-covering group of the elementary abelian p-group E of rank d. We also

have (L2V )/X ∼= (V ⊕ L2V )/(V ⊕X) in case (i) by Proposition 2.8.18. As V is an

irreducible G-module, with (L2V )/X or (V ⊕L2V )/X also an irreducible G-module

in case (i) or (ii), respectively, Theorems 3.3.9 and 3.3.10 imply that each p-group

PX or E∗/X has a unique proper nontrivial characteristic subgroup.

Theorems 3.2.10 and 3.3.9 show that, in each case, the p-group PX , E∗/X

or QX has the exponent-p class, nilpotency class and exponent of the p-group

given in Table 6.2.1. Here, we also use the fact that a p-group of exponent-

p class 2 has nilpotency class at most 2 and exponent at most p2, by Proposi-

tion 2.4.9. Furthermore, Corollary 2.11.9, Theorem 3.2.2 and Lemma 3.3.7 give

|PX | = pd(d+1)/2−dim(X), |E∗/X| = pd(d+3)/2−dim(X) and |QX | = pd(d+1)(2d+1)/6−dim(X),

with dim(X) = (d2−d)/2 in case (ii). In each case, the order of this p-group is given

in Table 6.2.1. The structure of A(P ) = N in cases (i) and (iii) is given by Lemmas

5.3.2, 5.4.9, 5.5.3 and 5.6.2. These lemmas also imply that N is a proper subgroup

of GL(d, p) in each case, and a proper subgroup of CSp(d, p) when G = E7(p).

We now show that the specified p-groups are optimal or quasi-optimal as re-

quired. Since the only p-group of rank d and exponent-p class 1 is E, with A(E) =

GL(d, p), we have that the optimal p-group in each case has exponent-p class at

least 2. Recall also from Proposition 2.4.9 that if a p-group has exponent p, then

its exponent-p class is equal to its nilpotency class. It follows immediately that the

p-group PX in case (i) is optimal with respect to N < GL(d, p). Additionally, The-

orem 6.2.1(ii) implies that, in case (ii), E∗/X is optimal with respect to G. Finally,

Theorem 6.2.1(i) shows that, in case (iii), QX is quasi-optimal with respect to N .

By definition, each p-group that is optimal with respect to N in this case has the


same exponent-p class, exponent and nilpotency class as QX .

In order to calculate the order of A(P ) in cases (i) and (iii) of the above theorem,

we must know the order of ZGL∩G. Proposition 5.1.2 and the discussion at the start

of each of Section 5.5 and Section 5.6 imply that this order is 3 when G = E6(p)

with p ≡ 1 (mod 3), and 2 when G = E7(p). Otherwise, G ∼= G is non-abelian and

simple, and hence ZGL ∩ G = 1.

The p-groups in cases (i), (ii) and (iii) of Theorem 6.2.2 can be constructed

as specific quotients of the universal p-group Γ(d, p, 2), the p-covering group E∗

of the elementary abelian p-group of rank d, and the universal p-group Γ(d, p, 3),

respectively, as detailed in the proof of the theorem. The p-groups in case (i) can also

be constructed as corresponding quotients of E∗, by Theorem 3.3.9. In addition,

Lemma 5.5.3 shows that if G = E6(p), then A(P ) is a C9-subgroup of GL(d, p).

Otherwise, A(P ) is not a maximal subgroup of GL(d, p), by Lemmas 5.3.2, 5.4.9

and 5.6.2. Therefore, the results of Theorem 6.2.2 are not covered by the work of

Bamberg et al. [7] summarised in Theorem 1.0.3. We also observe that the order of

each p-group in Table 6.2.1 is less than pd2, i.e., well below the value of pd

4/2 from

Theorem 1.0.3. Note that the information in the first row of Table 6.2.1, excluding

the general order of the optimal p-group, follows from our previous work [25].

Now, if Q is a p-group isomorphic to P , then A(P ) and A(Q) are the images of

representations afforded by isomorphic modules, with V the restriction of the former

module to G. However, our method of constructing an optimal or quasi-optimal p-

group in each case of Theorem 6.2.2 does not depend on the choice of the G-module

V . Therefore, if there are two isomorphism classes of minimal Fp[G]-modules, then

there are at least two isomorphism classes of optimal or quasi-optimal p-groups. By

Lemma 4.6.8, this is the case when G is equal to G2(3), or to E6(p) for some p > 3.

On the other hand, if M is a minimal Fp[G]-module isomorphic to V , then

M = V x for some x ∈ GL(V ), and it is easy to see from Propositions 2.11.6

and 2.11.7 that LrM = (LrV )x for each r ∈ 2, 3. In particular, since no two

submodules of LrV are equidimensional by Theorem 4.6.11, x maps the submodule

X of LrV from the proof of Theorem 6.2.2 to the unique submodule of LrV of

dimension dim(X), which we will denote by Y . It follows from Proposition 2.11.7,

Lemma 3.2.7 and the definitions of PX and QX that x induces an isomorphism

from PX to PY or from QX to QY , as appropriate. Therefore, in cases (i) and (iii)

of Theorem 6.2.2, there are exactly two isomorphism classes of optimal or quasi-

optimal p-groups, respectively, when G is equal to G2(3) or to E6(p), and exactly

one isomorphism class otherwise.

Chapter 7

Conclusion

Let q be a power of an odd prime p, let G be the simply connected version

of an exceptional Chevalley group G defined over Fq, and let G be the associated

linear algebraic group. In addition, let d be the dimension of a minimal Fp[G]-

module. We have determined the submodule structure of the exterior square of each

minimal Fq[G]-module, and of the exterior square of the corresponding irreducible

modules over Fp for G and for G. We have done the same for the third Lie power

of each irreducible module when G ∈ E6(q), E7(q) and p > 3, except in the case

of a few small values of p, where we have only determined the structure of the

third Lie power of each minimal Fp[G]-module. Conjecture 4.6.12 posits that the

submodule structure of a minimal Fp[G]-module V in a given “exceptional prime”

case is equivalent to that of the third Lie power of each corresponding irreducible

module defined over Fq or over Fp. It would be interesting to determine whether

this conjecture is true, and if not, then to determine the actual structures of the

other third Lie powers. As discussed at the end of Section 4.6, this would involve

determining in each case which composition series of L3V correspond to composition

series of the corresponding Fp[G]-module. For each G, we have also determined part

of the overgroup structure of G in GL(d, q). This information allowed us to identify

the stabiliser in GL(d, q) of each submodule of the aforementioned Fq[G]-modules.

If Conjecture 4.6.12 holds, then our results extend to the submodules of L3V when

p is an “exceptional prime” and q > p.

Next, let q = p and p > r, where r := 3 if G ∈ E6(p), E7(p) and r := 2

otherwise. Using our knowledge of the stabilisers of the aforementioned submodules,

we constructed a p-group P of exponent-p class r, nilpotency class r and exponent

p such that the group A(P ) induced by Aut(P ) on P/Φ(P ) is the normaliser of

G in GL(d, p). In the cases G ∈ G2(p), E8(p), we also constructed a p-group

P of exponent-p class 2, nilpotency class 2 and exponent p2 such that A(P ) =

G. The constructed p-group is optimal with respect to A(P ) in each case with

r = 2, and quasi-optimal when r = 3. Roughly, this means that P has the smallest

exponent-p class, exponent and nilpotency class of all p-groupsQ withA(Q) = A(P ),

and the smallest order when r = 2. However, it is possible that when r = 3,

there exists a p-group Q of exponent-p class 3, nilpotency class 3 and exponent

p such that |Q| < |P | and A(Q) = A(P ). If this is the case, then we would

135

136 Chapter 7. Conclusion

like to construct the smallest such p-group Q. By Proposition 3.2.5, Lemma 3.2.6

and Definition 3.2.11, this would involve studying the quotients of the universal p-

group Γ(d, p, 3) by normal subgroups that lie in γ2(Γ(d, p, 3)), and neither lie in nor

contain γ3(Γ(d, p, 3)). Additionally, in the discussion following our main theorem,

we determined the number of isomorphism classes of optimal or quasi-optimal p-

groups with respect to NGL(d,p)(G). It would be worthwhile to do the same for the

optimal p-groups with respect to G.

Now, in the cases where G ∈ G2(p), E8(p), we were able to induce precisely

G ∼= G on the Frattini quotient of a p-group because the exterior square of each

minimal module has a composition factor isomorphic to the minimal module. As we

saw in Section 4.5, the Lie algebra of G is a minimal K[G]-module when G = E8(q).

In fact, Sections 4.4 and 4.5 show that if G is an exceptional Chevalley group, or

if G ∈ 3D4(q), 2E6(q), and if U is the irreducible Fp[G]-module whose highest

weight is the highest weight of the Lie algebra of G, then U is isomorphic to a

composition factor of A2U . In almost all cases, U is the Lie algebra of G. In

the cases where G ∈ 3D4(q), 2E6(q) (with q odd), we have U = L(λ2), and the

graph automorphism of G of order 3 or 2, respectively, fixes U and each composition

factor of A2U [53, p. 180, p. 192]. We also see from Tables 4.5.10 and 4.5.14 that

the dimension of each of these irreducible modules is not an integral power of any

integer other than itself. Hence Proposition 4.6.6 implies that the corresponding

absolutely irreducible G-modules can be written over Fq, even when Fq is not a

splitting field for G. Furthermore, Corollary 4.2.3 and [56, Appendix A.2] imply

that if G 6∼= G (with G a Chevalley or twisted group), then Z(G) acts trivially on U .

In this case, since Z(G) 6 Z(G) [60, Corollary 24.13], and since G ∼= G/Z(G) is the

only nontrivial proper quotient of G by Proposition 4.6.9, the absolutely irreducible

Fq[G]-modules are actually faithful modules for G. Therefore, when q = p, it may

be possible to apply our methods to these modules, and the images of the afforded

representations in the relevant general linear groups, in order to construct a p-group

Q of exponent-p class 2 such that A(Q) is precisely G. Indeed, it can be shown that

several of the results in this thesis related to modules for a group H over a field

F, with the assumption that F is a splitting field for H, actually apply whenever

all relevant irreducible modules are absolutely irreducible. However, if G 6= E8(p),

then A(Q) would be a subgroup of a general linear group of dimension higher than

d, and hence the rank of Q would be greater than d.

Of course, Theorem 1.0.1 implies that, when G is an exceptional group of Lie

type, there is some p-group Q of rank d such that A(Q) is precisely the subgroup

G of GL(d, p). By Theorem 6.2.1, in order to construct such a p-group, we must

consider p-groups of a higher exponent or exponent-p class than those considered

in this thesis, or perhaps the aforementioned unexplored quotients of Γ(d, p, 3). It

would also be of interest to construct p-groups Q such that A(Q) is a group of Lie

137

type defined over a finite field whose order is not an odd prime (perhaps even a

Suzuki or Ree group). It may be possible to do this by applying our methods to

a subgroup of GL(n, p), for some n, that is isomorphic to such a group of Lie type

(possibly defined over a field of characteristic other than p). However, our current

methods of determining whether or not there exists a p-group Q of exponent p and

nilpotency class r ∈ 2, 3, such that A(Q) is a given subgroup of GL(n, p), apply

only when p > r. Developing corresponding methods for the cases with p 6 r would

allow us to study additional families of groups.

Finally, it would be worthwhile to investigate novel aspects of the general prob-

lem of inducing an arbitrary linear group on the Frattini quotient of a p-group.

Specifically, we may gain a deeper insight into the problem if we consider the fol-

lowing inverse of Problem 1.0.2:

Problem 7.0.1. Fix d > 1, and let P be the set of p-groups of rank d that satisfy

a certain fixed set of group theoretic properties. What can we say about the linear

groups H for which there exists P ∈ P with A(P ) = H?

For example, we may choose P to be the set of all p-groups (of rank d) of a given

exponent, nilpotency class and exponent-p class.

Appendix A

GAP and Magma code

In this appendix, we present the code that we used to perform the calculations

in the GAP [28] and Magma [11] computer algebra systems mentioned throughout

this thesis.

A.1 Properties of a 3-group in GAP

The following GAP code shows that the 3-group mentioned below the proof of

Proposition 2.4.9 has the specified properties. In particular, the presentation that

we gave for this group is easily derivable from the presentation returned by GAP.

G:=SmallGroup(3^6,24);

PClassPGroup(G) = 3; # true

NilpotencyClassOfGroup(G) = 2; # true

Exponent(G) = 9; # true

# Construct, simplify and display a presentation of G

T:=PresentationFpGroup(Image(IsomorphismFpGroup(G)));

TzGoGo(T);

TzPrintPresentation(T);

Note that we originally determined the existence of this group by calculating the

exponent-p class, exponent and nilpotency class of each group of order 36, which we

constructed using the SmallGroup command.

A.2 Exterior square non-isomorphism in Magma

Our first block of Magma code verifies that if V := F32, then V and A2V are

not isomorphic as F2[GL(V )]-modules, as we claimed below the proof of Proposition

2.8.24.

G:=GL(2,3); // G is isomorphic to GL(V)

V:=GModule(G);

A2V:=ExteriorSquare(V);

IsIsomorphic(V,A2V); // false

139

140 Appendix A. GAP and Magma code

A.3 Linear algebraic groups in Magma

Let G = Ym be a linear algebraic group associated with a simple group of Lie

type. We now describe the code used in the Magma calculations mentioned in

Sections 4.4 and 4.5, and also in the proof of Lemma 6.1.1. We first construct

several necessary objects in Magma.

R:=RootDatum("Ym");

W:=WeylGroup(GroupOfLieType(R,GF(5)));

A:=RootAction(W);

Phi:=Roots(R); // The roots of G

FW:=FundamentalWeights(R); // The fundamental dominant weights

For example, if G = E7, then the argument of the RootDatum command should

be "E7". Note that the specific field used to construct the Weyl group W does

not matter. Recall that the roots of G and the fundamental dominant weights of

a maximal torus of G are examples of characters in the character group of this

maximal torus. Each character λ that we construct in Magma is represented as a

tuple (c1, . . . , cm) of real numbers, where λ =∑i = 1mciαi, and where α1, . . . , αm

is a fixed base of the root system of G. If λ and µ are two such characters, then

λ < µ if and only if each entry in the tuple µ− λ is nonnegative, with at least one

entry positive.

The following function returns, as an enumerated sequence, the Weyl orbit of a

given character x.

function weylorbit(x);

return Setseq(Set(GSet(W,x,A)));

end function;

Recall from Proposition 4.2.15 that the Weyl orbit of a root of G consists of all

roots of G of the same length. The commands weylorbit(HighestShortRoot(R))

and weylorbit(HighestLongRoot(R)) can be used to calculate the short and long

roots of G, respectively, or to determine whether or not all roots of G lie a single

Weyl orbit.

Suppose, for example, that G = E7, and that V is the irreducible G-module

L(λ7) over Fp. Lubeck’s online data [58] shows that the weight multiset for V is

equal to the Weyl orbit of λ7. The following code calculates the weight multiset for

A2V and the weight multiset for X := (A2V ⊗ V )/A3V , as described by Lemmas

4.2.28 and 4.2.29.

WV:=weylorbit(FW[7]); // Weight multiset for V

WA2V:=[];

WA3V:=[];

for i in [1..#WV] do

for j in [i+1..#WV] do

A.3. Linear algebraic groups in Magma 141

Append(~WA2V,WV[i]+WV[j]);

for k in [j+1..#WV] do

Append(~WA3V,WV[i]+WV[j]+WV[k]);

end for;

end for;

end for;

WX:=[];

for i in [1..#WV] do

for j in [1..#WA2V] do

Append(~WX,WV[i]+WA2V[j]);

end for;

end for;

for i in [1..#WA3V] do

Exclude(~WX,WA3V[i]);

end for;

If G = C28 and if V is the 56-dimensional G-module L(λ1) defined over Fp, as in

the proof of Lemma 6.1.1, then Lubeck’s online data does not specify the weight

multiset for V . However, the command #weylorbit(FW[1]) shows that the Weyl

orbit of λ1 has size 56, and hence the weight multiset for V is exactly this Weyl

orbit, for the reason given in the proof of the aforementioned lemma.

Suppose that D is a disjoint union of weight multisets for composition factors

of A2V (or of X), stored in Magma as an enumerated sequence. We can use the

command Sort(~D) to sort the elements of D. The final weight D[#D] in the enu-

merated sequence is then a dominant weight λ such that no weight in D is higher

than λ. Thus L(λ) is a composition factor of A2V (or of X) by Lemma 4.2.37. We

can compare λ and the fundamental dominant weights stored in Magma in order

to express λ as a linear combination of these fundamental dominant weights, and

hence to determine the weight multiset for L(λ) via Lubeck’s online data. We can

then exclude the weights in this weight multiset from D in order to determine an ad-

ditional composition factor. Note that sorting the weight multiset for L(λ) reduces

the computation time here.

For example, if G = E7, and if D is the sorted weight multiset for A2(L(λ7)),

then the final element in D is λ6. If G is defined over Fp, with p /∈ 2, 7, then by

Lubeck’s online data, the weight multiset for L(λ6) consists of the Weyl orbit of λ6;

six copies of the Weyl orbit of λ1; and 27 copies of the weight 0. Using the following

code, we apply Lemma 4.2.37 to D, excluding from D the weights corresponding to

the composition factor L(λ6).

W1:=weylorbit(FW[1]);


W6:=weylorbit(FW[6]);

WL6:=W6;

for i in [1..6] do

WL6:=WL6 cat W1;

end for;

for i in [1..27] do

Append(~WL6,0);

end for;

for i in [1..#WL6] do

Exclude(~D,WL6[i]);

end for;

The smaller set D now consists of a single weight, 0. Thus the composition factors

of A2V are L(λ6) and L(0).

A.4 Submodule structures of Lie powers in Magma

In this section, we describe the Magma code mentioned in the proof of Theorem

4.6.11. Let q be a power of an odd prime p, let G = Ym(q) be an exceptional

Chevalley group, and let G be the simply connected version of G, as defined in

Lemma 4.2.1. If (G, p) 6= (F4(q), 3), then the group G and a minimal Fq[G]-module

V can be constructed as follows:

H:=ChevalleyGroup("Y",m,q);

V:=GModule(H);

For example, ChevalleyGroup("E",7,7) is the simply connected version of E7(7).

If (G, p) = (F4(q), 3), then we construct a minimal Fq[G]-module V using the fol-

lowing code.

H:=ChevalleyGroup("F",4,q);

U:=GModule(H); // Reducible 26-dimensional module

V:=Submodules(U)[2];

In each case, we construct A2V , which is isomorphic to L2V by Lemma 2.11.8, using

the command A2V:=ExteriorSquare(V). When q = p is an “exceptional prime” for

G, we can use the SubmoduleLattice(A2V) command to determine the submodule

structure of A2V . However, when G ∈ E7(p), E8(p), it is significantly faster to

determine this structure by using the MaximalSubmodules command recursively.

This calculation is longest when G = E8(5). Using the MaximalSubmodules method

A.4. Submodule structures of Lie powers in Magma 143

with a 2.6 GHz CPU, the calculation in this case completed after 4 CPU hours, and

the maximum RAM usage during this calculation was 7.9 GB.

Now, assume that p > 3, and let d := dim(V ). If e1, . . . , ed is a basis for V ,

then

(ei ∧ ej)⊗ ek + (ej ∧ ek)⊗ ei − (ei ∧ ek)⊗ ej | 1 6 i < j < k 6 d

is a basis for the submodule A3V of A2V ⊗ V [7, p. 2938]. We use the following

Magma code to construct the module L3V , which is isomorphic to (A2V ⊗ V )/A3V

by Lemma 2.11.8. Note that Magma orders its exterior square and tensor product

basis vectors in descending canonical order. For example, if U is a 3-dimensional

module with basis e1, e2, e3, then the ordered basis of A2U ⊗ U in Magma is

(e2 ∧ e3)⊗ e3, (e2 ∧ e3)⊗ e2, (e2 ∧ e3)⊗ e1,

(e1 ∧ e3)⊗ e3, (e1 ∧ e3)⊗ e2, (e1 ∧ e3)⊗ e1,

(e1 ∧ e2)⊗ e3, (e1 ∧ e2)⊗ e2, (e1 ∧ e2)⊗ e1.

A2VV:=TensorProduct(A2V,V);

// The basis vector for A3V in A2VV

// corresponding to the tuple (i,j,k)

function basisvec(i,j,k);

x:=d*(d*(i-1)-Integers()!(i*(i-1)/2))+d*(j-i-1)+k;

return A2VV.(Dimension(A2VV)-x+1);

end function;

basis:=[];

// Construct the basis for A3V in A2VV

for i in [1..d] do

for j in [i+1..d] do

for k in [j+1..d] do

Append(~basis,basisvec(i,j,k)+basisvec(j,k,i)\

-basisvec(i,k,j));

end for;

end for;

end for;

// The smallest submodule of A2VV containing all vectors in "basis"

A3V:=sub<A2VV|basis>;

L3V:=A2VV/A3V;


At this stage, we can delete the variables that we used to construct L3V , in order

to free up a significant amount of RAM, as follows:

delete V;

delete A2V;

delete A2VV;

delete basisvec;

delete basis;

delete A3V;

When G = E6(5), the submodule structure of L3V can be calculated using the

SubmoduleLattice command. However, it is faster to use the MaximalSubmodules

command recursively. In this case, the subset operator can be used to check that

if a submodule U of L3V contains a submodule X, and if another submodule W

of L3V contains a submodule Y with the same dimension as X, then Y = X.

For example, suppose that U , X and W are the submodules of L3V of dimension

5902, 5824 and 6474, respectively, from Figure 4.6.2. Assume also that we have

already checked that X is the unique 5824-dimensional submodule of U . Since the

submodule Y:=MaximalSubmodules(W)[2] of W has dimension 5824, and since Y

subset U evaluates to true, we have Y = X.

We now discuss our method of verifying the existence and containments of sub-

modules of L3V specified in the proof of Theorem 4.6.11, in the case where G = E7(p)

with p ∈ 7, 11, 19. Let R be a minimal F5[X]-module, where X is the simply con-

nected version of E7(5). Observe from Figure 4.6.3 that if n is an integer such that

an n-dimensional submodule of L3V is mentioned in the proof of Theorem 4.6.11,

then there exists a unique n-dimensional submodule Rn of L3R. In order to con-

struct a submodule of L3V of a given dimension n, we use Magma to search for a

vector r ∈ Rn such that:

(i) r is an element of the basis for Rn stored in Magma;

(ii) Rn is the smallest submodule of L3R that contains r;

(iii) if r is expressed as a linear combination of basis vectors for L3R, then the

coefficient of each basis vector is either 0, 1 or −1; and

(iv) the number of nonzero coefficients in this linear combination is as small as

possible.

Here, (iii) allows us to use exactly the same linear combination of basis vectors (in

terms of coefficients) in order to construct the n-dimensional submodule of L3V ,

while the purpose of (iv) is aesthetic.

In order to calculate the vector r, we construct Rn and its maximal submodules

by using the MaximalSubmodules command recursively. For a given basis vector u of


Rn, the in operator can be used to check if u lies in no maximal submodule of Rn, i.e.,

if u satisfies (ii). The command c:=Coordinates(L3R,u) can be used to determine

the coefficients of the associated linear combination of basis vectors for L3R, and

Index(c,x) eq 0 evaluates to true for each x ∈ 2, 3 if and only if u satisfies (iii).

Additionally, we obtain the number of nonzero coefficients in the linear combination

using the command NumberOfNonZeroEntries(Matrix(GF(5),1,58520,c)). Us-

ing a 2.6 GHz CPU, our computations here completed after 49 CPU hours, with a

maximum RAM usage of 31.8 GB.

Once we have calculated r, we extract each nonzero coefficient in the associated

linear combination, as well as the coordinate of the corresponding basis vector,

i.e., its position in the ordered basis for L3R. We then use this data to construct

the vector v ∈ L3V whose linear combination of basis vectors is the same as r.

Subsequently, we construct the smallest submodule Un of L3V that contains v, and

we verify that this submodule has dimension n. We use the following function

to construct v and Un, where coeffs is the enumerated sequence of the nonzero

coefficients in the linear combination for r, and where coords is the enumerated

sequence of the coordinates of the corresponding basis vectors.

function smallestsubmodule(coords,coeffs);

v:=0*L3V.1;

for i in [1..#coords] do

v:=v+coeffs[i]*L3V.coords[i];

end for;

return sub<L3V|v>;

end function;

We now present the code that we used to verify the necessary information about

L3V with G = E7(7).

// Construct the 7392-dimensional submodule

coords:=[3099, 3118, 3137, 3157, 3177, 3197, 3218, 3261, 3304, 3347,\

3412, 3477];

coeffs:=[1, -1, 1, -1, 1, -1, -1, 1, -1, 1, -1, 1];

U7392:=smallestsubmodule(coords,coeffs);

Dimension(U7392) eq 7392; // true

M:=MaximalSubmodules(U7392);

#M eq 1; // true

Dimension(M[1]) eq 6480; // true

N:=MaximalSubmodules(M[1]);

#N eq 1; // true

Dimension(N[1]) eq 912; // true


// Free up some RAM

delete U7392;

delete M;

delete N;


coords:=[8092, 8901, 9766, 10689, 11672, 12717, 13826, 15001,\

16244, 17557, 18942, 20401, 21936, 23549, 25242, 27017,\

28876, 30821, 32854, 34977, 37192, 39501, 41906, 44409,\

47012, 49717, 52526, 55441];

coeffs:=[1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1,\

1, 1, -1, -1, 1, 1, -1, 1, -1, 1, -1];

Dimension(smallestsubmodule(coords,coeffs)) eq 56; // true

// Construct the 51072-dimensional submodule,

// where coords = coeffs = [1]

Dimension(sub<L3V|L3V.1>) eq 51072; //true

The following code was used to perform the necessary calculations with G =

E7(11).

// Construct the irreducible submodules of L3V

M:=MinimalSubmodules(L3V);

#M eq 3; // true




U6480:=M[3];

// Free up some RAM

delete M;


// where coords = [76] and coeffs = [1]

U57552:=sub<L3V|L3V.76>;


W:=U57552/U6480;

// Free up some RAM

delete U57552;


delete U6480;

N:=MinimalSubmodules(W);

#N eq 1; // true


Finally, we present the code corresponding to the case where G = E7(19).


coords:=[44460, 44664, 44715, 44919, 45072, 45276, 45378, 45480,\

45531, 45684, 45735, 45888, 45990, 46092, 46143, 46194,\

46296, 46347, 46449, 46500, 46551, 46704, 46755, 46806,\

49559, 52365s, 55277, 58297];

coeffs:=[1, -1, 1, 1, -1, 1, -1, -1, 1, -1, -1, 1, -1, -1, 1, 1,\

-1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, -1];

U6536:=smallestsubmodule(coords,coeffs);


M:=MaximalSubmodules(U6536);

#M eq 1; // true


N:=MaximalSubmodules(M[1]);

#N eq 1; // true


// Free up some RAM

delete U6536;

delete M;

delete N;


coords:=[43434, 43586, 46016, 46171, 46832, 48700, 48858, 49531,\

49582, 51488, 51648, 52334, 52386, 52437, 54381, 54544,\

55243, 55295, 55347, 55399, 57382, 57547];

coeffs:=[1, -1, -1, 1, 1, 1, -1, -1, 1, -1, 1, 1, -1, 1, 1, -1, -1,\

1, -1, 1, -1, 1];

Dimension(smallestsubmodule(coords,coeffs)) eq 912; // true


// where coords = coeffs = [1]

Dimension(sub<L3V|L3V.1>) eq 51072; // true


These computations (including the construction of L3V but excluding the con-

struction of the vectors r ∈ L3R), which we again ran using a 2.6 GHz CPU,

completed after 0.6 CPU hours in the case of E7(7); 223 CPU hours in the case of

E7(11); and 4 CPU hours in the case of E7(19). The maximum RAM usage during

the computations for each of these three cases was 14.5 GB, 67.0 GB and 38.7 GB,

respectively. The long calculation time and high memory usage in the case of E7(11)

are due to the calculations of irreducible submodules of high-dimensional modules.

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Index

Albert algebra, 115

algebra, 20

almost simple group, 9

Aschbacher class, 37

base of a root system, 61

C9-subgroup, 37

character group, 59

character of a linear algebraic group,

59

dominant, 62

restricted, 66

Chevalley group, 55

classical group, 17

commutator, 5

Dedekind’s Identity, 5

derived series, 7

derived subgroup, 5

elementary abelian group, 11

equivalent F-representations, 22

exponent of a group, 7

exponent-p class, 12

exterior power, 20

exterior square, 20

F-representation

absolutely irreducible, 33

F[G]-module

absolutely irreducible, 33

dual, 23

faithful, 22

minimal, 22

multiplicity free, 24

semisimple, 24

trivial irreducible, 21

twisted by an automorphism, 23

uniserial, 24

written over a subfield, 33

field automorphism, 63

form, 14

alternating, 14

bilinear, 14

degenerate, 15

group of isometries, 15

group of similarities, 15

σ-Hermitian, 14

matrix, 16

minus type, 16

non-degenerate, 15

orthogonal, 14

orthogonal complement, 15

plus type, 16

preservation, 15

quadratic, 14

reflexive, 14

symmetric, 14

symplectic, 14

Frattini subgroup, 8

free Burnside group, 9

general orthogonal group, 18

generally quasisimple classical group,

18

geometric subgroups, 36

group of Lie type

157

158 index

exceptional, 55

simply connected version, 57

twisted, 55

Hermitian matrix, 115

Jacobi identity, 38

Lie algebra, 38

of a linear algebraic group, 59

Lie bracket, 39

Lie power, 39

lower central series, 10

lower exponent-p central series, 12

nilpotent group, 10

octonion algebra, 109

optimal p-group, 42

p-covering group, 49

p-group, 11

p-multiplicator, 49

perfect group, 6

projective group, 19

quasi-optimal p-group, 49

rank of a group, 8

rank of a linear algebraic group, 59

Ree group, 55

root of a linear algebraic group, 60

long, 61

short, 61

root system, 60

Schur multiplier, 10

simple group of Lie type, 55

special orthogonal group, 18

special unitary group, 18

splitting field, 34

Suzuki group, 55

symplectic group, 17

tensor algebra, 21

tensor power, 20

Theorem

Aschbacher’s, 38

Burnside’s Basis, 12

Clifford’s, 31

Steinberg’s Tensor Product, 66

torus, 59

maximal, 59

universal p-group

rank d and exponent-p class 2, 51

rank d, exponent p and nilpotency

class r, 43

universal cover, 9

wedge product, 20

weight

fundamental dominant, 61

highest, 64

of a module, 62

weight space, 62

Weyl group, 60

Weyl module, 67

Documents

-groups related to exceptional groups of Lie type€¦ · p-groups related to exceptional groups of Lie type Saul D. Freedman, BSc (Hons), BE (Hons) October 2018 Department of Mathematics