Maximal Subgroups of Finite Groups - University of Virginiapi.math.virginia.edu/~lls2l/maximal-subgroups-of-finite-groups.pdf · MAXIMAL SUBGROUPS OF FINITE GROUPS (I) Determine ~t.G~HI(G,V)

JOURNAL OF ALGEBRA 92, 44-80 (1985)

Maximal Subgroups of Finite Groups

M. ASCHBACHER AND L. SCOTI'*,t

Department of Mathematics, California Institute of Technology,Pasadena, California 91125 and*Department of Mathematics,

University of Virginia, Charlottesville, Virginia 22903

Communicated by Walter Feit

Received February 15, 1983

What ingredients are necessary to describe all maximal subgroups of thegeneral finite group G? This paper is concerned with providing such ananalysis.

A good first reduction is to take into account the first isomorphismtheorem, which tells us that the maximal subgroups containing a givennormal subgroup N of G correspond, under the natural projection, to themaximal subgroups of the quotient group GIN. Let n = nG denote thecollection of maximal subgroups of G, and let n * be the subset of thoseMEn with KerG(M) = 1, where KerG(M) denotes the largest normalsubgroup of G contained in M. Then the first isomorphism theorem allows usto identify n with the disjoint union UN <lG nS/N' Actually, what we reallywant to parameterize are the conjugacy classes of maximal subgroups, butthis too works well: If W = WG denotes the set of G-conjugacy classes ofelements of n, and W* is defined similarly, then we have

WG~ U WS/N'N<lG

Hence our analysis is reduced to n* and W*, rather than nand W, if weassume a knowledge of the normal subgroup structure of G. The reduction isreally even better than that, since often #zG/N = ¢i. For example, if G is a pgroup, then nS/N = ¢i unless IGIN I = p.

Our main result is Theorem 1 below, which gives a general structuraldescription of finite groups G with nS nonempty, describes what tl~ lookslike, and determines WS up to some difficult but well-defined problems:

Let G be a finite group with L <G <Aut(L) for some nonabelian simplegroup L, and let V be faithful irreducible G-module over some field ofprimeorder.

t Both authors receive partial support from the National Science Foundation.

440021-8693/85 $3.00Copyright © 1985 by Academic Press. Inc.All rights of reproduction in any form reserved.

MAXIMAL SUBGROUPS OF FINITE GROUPS

(I) Determine ~t.G~HI(G, V).

(2) Determine ~~ .

45

It is likely that a considerable knowledge of the irreducible modules forsimple groups will enter into any final solution of either problem, cf. [4].(For some corrections to [4] see the Appendix to this paper.)

The first cohomology group H1(G, V) of a group G on a G-module V hasmany interpretations, but the relevant one here is as the set of conjugacyclasses of complements to V in the semidirect product V.G. This interpretation carries over to the case where V is nonabelian (cf. the discussion inSection 2) though H1(G, V) is no longer a group. In the process of obtainingTheorem I we also prove a result on nonabelian I-cohomology ofindependent interest; that result is recorded in Theorem 2. Among otherthings, Theorem 2 gives the existence of some unusual subgroups in wreathproducts that are more or less invisible to standard techniques in grouptheory. Cf. the discussion of part (C)(I) of Theorem I at the end of the nextsection.

Finally, if G is a finite group and V a faithful irreducible module for Gover a field of prime characteristic with H1(G, V) =t- 0, then in Theorem 3 wedetermine the structure of the generalized Fitting subgroup of G and therepresentation of that subgroup on V.

STATEMENT OF THE MAIN RESULTS

We continue the notation introduced above with G a finite group. Aut(G),Inn(G), and Out(G) denote the group of automorphisms, innerautomorphisms, and outer automorphisms of G, respectively. If H/K is asection of G then AutG(H/K) denotes the group of automorphisms of H/Kinduced in G; thus

AutG(H/K) ~ (NG(H) n NG(K))/CG(H/K)

and CG(H/K) consists of those g E NG(H) n NG(K) with [H, g] <. K.OutG(H/K) is the image of AutG(H/K) in Out(H/K).

We will often consider a group D with a given direct product decomposition D = nleIL;. A diagonal subgroup A of D (with respect to thisdecomposition) is a subgroup for which each projection A -4 L; is injective; ifthese maps are in fact all isomorphisms then A is afull diagonal subgroup ofD. .Y; denotes the set of full diagonal subgroups of D and @D denotes the Dconjugacy classes of such subgroups. For J r::;;. I write DJ = fLeJ L j andabbreviate @D

J= @J' etc., with Y =.Y; and @ = @D when no confusion can

arise. Suppose G acts on D and permutes L1 = {L;: i E I}. Then we denote by@(G) the G-stable classes in @. Let 3'(G) denote the G-stable partitions r

481/92/1-4

46 ASCHBACHER AND SCOTT

of .1 and .9'*(G) the maximal nontrivial G-stable partitions of .1; that is,r GE .9'*(G) ifr' E .9'(G) and r is minimal subject to this constraint and toIrl>!. Notice we do allow r=LJ. For LELJ, write .9'*(G,L) for thecollection of subsets r of .1 with L E rand r' E .9'* (G).

Recall the generalized Fitting subgroup of G is the subgroup F*(G)generated by all subnormal nilpotent or quasisimple subgroups of G, with thelatter subgroups called the components of G. A group is quasisimple if it isperfect and simple modulo its center. It turns out F*(G) is the centralproduct of the Fitting subgroup of G and the components of G. Animportant special case of our analysis occurs when D = F*(G) is the directproduct of the set .1 of all G-conjugates of some simple component L of G;in that event the notation and terminology of the last paragraph applies, andis used without comment.

We are now ready to state the main theorem.

THEOREM 1. Let G be a finite group with n* nonempty and setD = F*(G). Then one of the following holds:

(A) D is an elementary abelian p-group for some prime p, G actsirreducibly on D, and n* is the set of complements to D in G. If I<?f *I > 1then F*(GID) is the direct product of the conjugates of a simple componentL of GID, CdL) centralizes u= [D,L], AutG(L) acts faithfully andirreducibly on U, and there is a natural bijection

<?f* ~ H1(AutdL), U).

(8) D is the direct product of the G-conjugates of L X K where LandK are isomorphic nonconjugate simple components of G. Let ~ (L) consist ofthose U in KG with NG(L) = NdU) and Dw(NG(L» =t- 0. Then n* consistsof those subgroups M of G such that G = MD and M n D is the directproduct of the M-conjugates of a full diagonal subgroup of L U(M) for someU(M) in 'l/(L). The map M G H (M n L U(M»D gives a bijection

<?f* ~ U @w(NG(LU».UE it'lL)

Indeed <?f* ~ UUEit'(l.) COUI(L)(OutG(L».

(C) D is the direct product of the set .1 of G-conjugates of some simplecomponent L of G. The set n* is the disjoint union of G-stable subsets nt,1 ~i ~ 3, corresponding to the three cases below, and <?f * decomposesaccordingly. Specifically

ni = {N E n*: AutN(L) = AutdL) and N n D = I}

ni = {N E n*: AutN(L) = AutdL) and N n D =t- 1}

nJ = {NE n*: AutN(L) is maximal in AutdL)}

leading to the following three cases:

MAXIMAL SUBGROUPS OF FINITE GROUPS 47

(1) Let n; consist of those complements M to D with Inn(L) ~AutM(L). Then n: consists of those members of n; contained in no memberof nt. Moreover there is a natural bijection between the collection W; oforbits of G on n; and the set of D-classes of complements X to DjCD(L) inNG(L)jCD(L) with Inn(L) ~ Autx(L).

Theorem 4 gives a necessary and sufficient condition for a member ofn; tobe contained in n: and gives a parametrization of W't-

(2) nt consists of those subgroups M of G such that G = MD, M n Dis the direct product of the M-conjugates of M n Dr E~, for some r withr E ,:?*(G). The map M GH (M n Dr)D r gives a bijection

Wr ~ U 92n /NG (D r ))·rE,'f"(G.L)

Theorem 5, applied to the various Dr' makes this parametrization explicit.

(3) n3* consists of those subgroups M of G such that G = MD, M n Dis the direct product of the M-conjugates of M n L, AutM(L) is a maximalsubgroup of AutG(L) which does not contain Inn(L), and AutM(L) nInn(L) = AutMnL (L). The map M G---> (AutM(L ))AUIG(L) gives a bijection

=*~=*70 3 = 70 AUIG(L)'

THEOREM 2. Let X be a group containing a normal subgroup D which isthe direct product of the X-conjugates of some subgroup L. LetD' = (L x - {L}) and if T is a complement to D in X define 1i(T) =D'N T(L)jD '. Then Ii is a surjective map from the set of all complements to Din X onto the set ofall complements to DjD' ~ Lin Nx(L)jD', and Ii inducesa bijection

of conjugacy classes of complements.An auxiliary result (3.6) also describes which complements to Din Nx(L)

have the form N T(L) for some complement T to D in X.

It was recently pointed out to us that on the one hand there is someoverlap between Theorems 1 and 2 and some work of Gross and Kovacs in[7,8], and on the other that Proposition 1.29 in [61 is the split case ofTheorem 2.

THEOREM 3. Let G be a finite group, p a prime, K a field of characteristic p, and V a faithful irreducible KG-module such that HI (G, V) =I=- O.Then

(1) F*(G) is the direct product of the G-conjugates of a simplecomponent L of G o}"Order divisible by p.


(2) V is the direct sum of the G-conjugates of U = [V, L 1.(3) NG(L) acts irreducibly on U with CG(L) = Cdu).

(4) H1(G, V) ~ HI (AutdL), U).

(5) dim(H1(G, V))~dim(HI(L,E)) for each nontrivial irreducibleKL-submodule E of V.

THEOREM 4. Assume the hypothesis and notation of Theorem 1, andassume case C of Theorem 1 holds. Let g be the set of normal subgroups Eof NdL) such that

(i) D~E~LCG(L) with DCdL)jE~L, and

(ii) @«CG(Ll/CE(L))XLl(NG(L)) is nonempty, and

(iii) for each rE5'*(G,L) with @n/NG(Dr)) nonempty, thefollowing two conditions hold:

(a) either E or DCG(L) is not normal in Nr;(Dr ), and

(b) the section NdL)jE has no normal complement in NdDr ).

Then we have

(1) n i consists of those MEn: such that DCM(L) E g.

(2) 'dfi ~ g X Cout(L)(OutG(L)).

By definition if A <J B ~ C are groups then a complement to the sectionA jB in C is a subgroup N of C with BN = C and B n N = A.

Lemma 7.30 gives another parameterization of'dfi in terms of NG(L)invariant L-orbits of homomorphisms from DCG(L )jD to L. SimilarlyTheorem 5 in Section 5 gives a bijection between @(G) and the set ofhomomorphisms from G into Out(L) extending the conjugation map fromNG(L) to Out(L) (in the event @(G) is nonempty). The latter result showsl'dfil ~ 15'*(G,L)IIOut(L)I, for example (cf. 5.15 or 6.4.3.)

The question of when 'df* is nonempty in case A or B and when 'dfi isnonempty in case C are left open. The first question simply asks if a certaingroup extension splits, and the other two have a similar flavor.

The maximal subgroups in part (C)(l) seem particularly interesting; eachis a complement to D = F*(G) in G, although very complement need not bemaximal. Theorem 2 gives a bijection between the complements to D in Gand the complements to L in NG(L)jCn(L), and Theorem 4 and Lemma 7.30supply a means for deciding maximality. Lemmas 7.1 and 7.2 give ad hocmethods for deciding whether 'dfi is nonempty. To illustrate the theory,consider the wreath product G = As wr A 6 • Then NG(L)jCn(L) ~As X As, soa full diagonal subgroup X of NG(L)jCn(L) is a complement to L. Hence byTheorem 2 there exists a complement M to D in G with NM(L) CD(L)j


CD(L) =X, and by 7.2 M is maximal in G. Carlos Scoppola supplies analternate argument for the existence of at least one nonstandard complementin G = As wr A 6 • Namely, A 6 has a transitive representation on 30 letterswith 6 blocks of imprimitivity of size 5, and hence is embedded in a copy ofG in S 30 preserving this set of blocks. I

1. DIAGONAL SUBGROUPS

Let L be a group, ff = (a;: L -+ L;: i E I) a family of isomorphisms, andD = TI;E[L; the direct product of the groups L;, i E I. For x E L define thediagonal of .~ to be

diag(ff) = \nxa;: x ELI.IIE[ \

Let Aut[(D) be the subgroup of Aut(D) permuting Ll = {L;: i E l}.

(1.1) Let S be the symmetric group on I, L t = Aut(L;), andD* = TI;E[L;*. a; induces the isomorphism

at: L* = Aut(L) -+ Lt

and if we set ff* = (at: i E I) and let S act on D via

then

s: n x;a; -+ n x;sa;,iEI iEI

sE S,x;EL

(1) A = Aut[(D) is the semidirect product of D* and S.

(2) S acts on D* via

S • n * * n * *. x; a; -+ x;sa;,iEI iEI

SES,x;*EL*.

(3) NAdiag(.T) = S X diag(ff*) ~ S X Aut(L).

Proof A d is isomorphic to a subgroup of D *, while D * ~ Ad' soD* =Ad <J A. Thus D*S is subgroup of A. Also Ad ~ Sym(Ll) = Sd, soAd = Sd. Hence A = SA d = SD *, and as Sd = 1, the product is semidirect.So (1) holds.

For i E I let D; = T1*; L;. Then Lt = CD.(D;) so for s E S, (Lty = L;~,

and in particular V = Ns(L;) = Ns(Lt). Further [V, Lt] ~ CA(L;) nCA(D/) = CA(D) = 1. Therefore (2) holds.

I Note added in proof K. Gruenberg and L. Kovacs have informed the authors that theyindependently obtained a result similar to Theorem 3.


Let B = diag(Y) and B * = diag(Y*). Evidently B *S = B * X S ~

NA(B) = M. MtJ. ~AtJ. = StJ., so M = SMtJ.. B* ~ MtJ. with Aut(B) ~ B* andB* is faithful on B, so MtJ.=B*C(B)tJ.. Finally, if gEC(B)tJ. then gcentralizes xa and hence also its projection xa j on L j for each x ELandi E I. Thus g E CA(D) = 1, so C(B)tJ. = 1 and M = SB*.

Remark. In Section 3 we discuss wreath products. Notice that from thatdiscussion and Lemma 1.1, Aut[(D) is just the wreath product L * wr[ Swhere S is the symmetric group on I and L * = Aut(L ).

A diagonal subgroup of D is a subgroup X of D such that each projection7r j : X -+ L j, i E I, is an injection. A full diagonal subgroup is a diagonalsubgroup for which each projection is an isomorphism.

(1.2) Let if E I, L = L j" qj) the set offull diagonal subgroups of D, andJ?I' the set offamilies Y = (a j:L -+ L j: i E I) of isomorphisms with a/ = 1.Then

(1) The map Y -+ diag(Y) is a bijection of J?I' with ffJ.

(2) The map in (1) commutes with the actions of CD.(L) on ffJ and onJ?I' by right multiplication. D * is transitive on ffJ and CD.(L) is regular onffJ.

(3) If L and I are finite then IffJl = IAut(L)ln, where III = n + 1.

Proof For X E ffJ define aJ = 7ri,17rj: L -+ L j and Yf(X) = (aJ: i E I).Then Y'(X) E J?I' with X = diag(Yf(X». Conversely if Y E ffJ thenY'(diag(Y» =Y, so (1) holds.

Evidently the map in (1) commutes with the actions of CD.(L), andCD.(L) is regular on J?I', so (2) holds. Assume L and I are finite withIII = n + 1. As CD.(L) is regular on J?I', IffJ 1= 1 J?I' I = ICD.(L)I = 1 Aut(L )In,so (3) holds.

(1.3) Let G~ Aut[(D) with G transitive on .,1. Let X ~ L = L j, withXNG(L) = XL. Let n be the set of subgroups Y ofD such that Y = DjE/ XJorsome X j E X Gn L j' Then D is transitive on n if I and L are finite.

Proof Let (9 = XL. As (9 is N G(L )-invariant, (9G is of ordern=IG:NG(L)I=IJI. Hence

for some XjEXGnL j • Now Inl=I(9ln =IL:NL (X)ln=ID:ND (Y)I forY = DjE[Xj E n. So D is transitive on n.

In the remainder of this section we assume L to be a finite nonabeliansimple group and assume I is finite. Then we may identify L j with its groupInn(LJ of inner automorphisms and hence identify D with the normal


subgroup Inn(D) of A = Aut{(D). Moreover Aut(D) permutes thecomponents Lj> i E I, of D, so A = Aut(D)_

Let g be the set of all partitions of Ll. P E.9' is nontrivial if Irl > 1 forsome rEP. If G";; A then G permutes .9' in an obvious fashion. Partiallyorder g by P";; Q if P is a refinement of Q. The minimal nontrivial Ginvariant partitions (mentioned in the statement of Theorem 1) are theminimal members of the set of all nontrivial G-invariant partitions under thispartial order.

For r s; Ll let Dr = OKer K. Let !Jj be the set of tuples (Hr :rEP) as Pvaries over .9' and H r varies over the set of full diagonal subgroups of DrPartially order ,9J by (Hr:rEP),,;;(Ii(J:fJEP) if P";;P and H r is a fulldiagonal subgroup of O(Jep(r) Ii (with respect to that decomposition), wherepen consists of those members of P contained in r.

Let Jr be the collection of subgroups H of D such that the projectionsn;: H -4 L; are surjections for each i E I. Partially order Jr by inclusion.

We now state two lemmas from [4]. For completeness we include proofsof these lemmas.

(1.4) Let HE Jr. Then there exists PEg such that H = Orep Hnr isthe direct product of the full diagonal subgroups Hnr of Dr' wherenr : H -4 Dr is the projection map with respect to the decompositionD = IlreP Dr'

Proof. Let r s; Ll be minimal subject to K = H n B 1= 1, where B = DrBy minimality of r the projection n;: K -4 L; is nontrivial for each L j E r. AsB<lD, K=HnB<lH, so I1=Kn j <lHn j =L j _ As L j is simple weconclude n;: K -4 L j is a surjection for each L; E r. Next ker(n;)";;OJer-1L;1 J, so by minimality of r, ker(n;) = 1. Thus n j is an isomorphismand K is a full diagonal subgroup of B.

Let E = Dd -r, so that D = EX B. Let n: H -4 B be the projection withrespect to this decomposition. Again K = Kn <l Hn. But as K is a fulldiagonal subgroup of B, 1.2 and 1.1.3 imply K = NB(K), so Hn = K. ThusH = K X CH(K) with CH(K) = ker(n) = H n E. Finally, for L; E Ll - r,L j = Hn; = (H n E) n i , so the lemma holds by induction on the order of I.

(1.5) (1) The map (Hr:rEP)-4 OrePHr is an isomorphism of ,9Jand Jr as partially ordered sets.

(2) If HEJr and H";;X";;D then XEJr.

Proof If HEJr and H";; X";; D, then L;=Hn;";;Xn;, so XEJr and(2) holds.

Let qJ be the map defined in (1). Evidently qJ maps !Jj into Jr andpreserves the partial order. By 1.4, qJ is a surjection. As the components ofOrep H r are the groups H r , rEP, qJ is an injection. Suppose H = Orep H r

52 ASCHBACHER AND scon

is a subgroup of TIo EP ilO' Let 7C o : H --+ il0 and 7C;: D --+ L;, i E n be theprojections. L; = H7C; = H7C o 7C; < (ilo ) 7C; and 7C j : Ho--+L; is an isomorphism,so 7Co is a surjection. Thus by 1.4 there is a partition pi of P such that foreach rEP, H r is a full diagonal subgroup of TIo E II il0 for some eE P'.Then r = UOEII n, so P is a refinement of P and (Hr= rEP) < (ilo : n E P)as desired.

(1.6) Assume G <A = Aut(D) and M <G with G = MD, M is transitiveon A, and H = MnD = nrEP H r EJr,for some (Hr :rE P) Egj. Then

(1) M = NG(H).

(2) P is G-invariant.

(3) For each rE P, (Hrt r is a NG(Dr)-invariant class of fulldiagonal subgroups of Dr'

(4) Assume P is a G-invariant refinement of P and for n E P let il0

be the projection of H on D f1' Then il = TIo EP il0 is an M-invariantmember ofJr containing Hand (ilot n is NG(Do)-invariant.

(5) ME nG if and only if P is a minimal nontrivial G-invariantpartition of A.

Proof As D is trivial on A and G = MD, P E ,'j' is G-invariant preciselywhen P is M-invariant. As Hf> rEP, are the components of HandH = M n D <l M, P is M-invariant. Thus (2) holds. Also N G(H) = MNn(H)and Nn(H)=TIrEP(N(Hr)nDr). By 1.2 and 1.1.3, Hr=N(Hr)nDr , so(1) holds. Also for rEP, NG(Dr ) = DNM(Dr ) and NM(Dr ) <N(Hr ). Hence(Hr)n r is NG(Dr)-invariant, so (3) holds.

Assume the hypothesis and notation of (4). For n E P, N M(D 0) acts onthe projection H o of H on Do and for gEM, ngEP, and ilog=(Ho)g.Hence M <N G(H). Of course H <il, so mil n D = H and the last part of(4) follows from (3).

If MEn then by (4), P is a minimal nontrivial G-invariant partition of A.Conversely assume P is maximal and let M < Y <G. Then Y = MX, whereH <X = Y n D. By 1.5, X = nOEP X 0 for some refinement P of P. AsM <N(X), P is G-invariant, so by minimality of P, P is trivial. Hence X = Dand Y=MX=G. That is, MEn.

2. I-COHOMOLOGY

In this section we recall some facts about I-cohomology.Let G be a group acting on a group A. In this section we write our actions

on the left. For x, y E G, xy = yX- 1 = xyx- 1• A.G denotes the semidirect


product of A and G. The set reG, A) of cocycles consists of the functionsy: G -> A satisfying the cocycle condition

for all g, h E G. Yg denotes the image of g in A under y, and for a E A, ga isthe image of a under g.

Given a cocycle y, let y*:G->A.G be defined by y*(g)=ygg. Thecocycle condition is equivalent to the assertion that y* is a homomorphism.Indeed the map y-> y*(G) is a bijection of r(G,A) with the set ofcomplements to A in A.G. Moreover if a E A, then a(y*(G)) = fJ*(G), wherefJ is the cycle

Cocycles y and y' are defined to be cohomologous if there exists a E A withy; = a y ga -I for all g in G. Cohomology is an equivalence relation and wewrite lY] for the cohomology class of y, and write HI(G, A) for the set ofcohomology classes. From the discussion above the map [y] -> Ay*(G)defines a bijection between HI(G,A) and the set of conjugacy classes ofcomplements to A in A .G. The trivial class in HI (G, A) is the class of thetrivial cocycle mapping each element of G to 1. This cocycle corresponds tothe "standard copy" of G in A.G.

(2.1) JfG' isacomplementtoA inA.GthenH1(G,A);:;HI(G',A). Thisassertion is immediate from the bijections of HI (G, A) and HI (G', A) withthe classes of complements to A in A.G.

A acts on Hom(G, A) via:

a EA, g E G, qJ E Hom(G, A).

Denote by%om(G,A) the set oforbits ofA on Hom(G,A) under this action.

(2.2) Assume G acts by inner automorphisms on A in the sense that thereexists i E Hom(G, A) with ga = i(g)a for each a E A, g E G. For y E T(G, A)define

n(y): -> A

gH yg' i(g).

Then n(y)E Hom(G,A) and the map

[y] H A(n(y))

defines a bijection HI(G,A);:;Jrom(G,A).


Proof. A straightforward calculation shows n(y) E Hom(G, A) and themap [y1----' A(n(y» is a well-defined injection. If U E Hom(G,A) then definey: G ----. A by yg = u(g) i(g)-'. Then y E r(G, A), so the map is a surjection.

Some more notation: Let drom*(G,A) denote the set of orbits ofsurjective homomorphisms. If Z is a group acting on G and A then Z acts onHom(G,A) via ('rp)(g)=Z(rp(gZ», zEZ, gEG, rpEHom(G,A). Furtherthis action induces an action of Z on drom *(G, A) defined by Z(rpA) = ('rp)A,Z E Z, rpA E drom(G, A). Denote by dromt(G, A) the fixed points of Z on,?,om *(G, A) under this action.

(2.3) Let A be a nonabelian simple group normal in G and let .# denotethe set of conjugacy classes M G of complements to A in G such thatInn(A) <AutM(A). For M GE.# define flM to be the homomorphism fromCG(A) to A mapping an element to the inverse of its projection on A withrespect to the decomposition G = A .M. Then the map fl: M G----. (flM ).1 definesa bijection.# ~dromJ(CG(A),A).

Proof. Let K = CG(A) and notice Hom(K, A), drom(K, A) = H'(K, A),and drom *(K, A) consists of the orbits rpA such that Inn(A) = Aut""(K)(A).Moreover the map M G H (M n KA)G defines a bijection between s¥ and theG-invariant classes of complements rp*(K)A with rpA E drom *(K, A). Henceto complete the proof we must show rpA E ,Jrom*(K, A) is a fixed point of Gif and only if rp*(K)A is G-invariant.

If rp*(K)A is G-invariant then G = NG(rp*(K»)A, and NG(rp*(K» commuteswith the map rp, so that rpA is a fixed point of G. Conversely if rpA is a fixedpoint, then as the action of A on Hom(K, A) is the restriction to A of theaction of G on Hom(K, A), G = AM, where M is the stabilizer of rp in G, andrp*(K) = M n KA <l M, so that rp*(K)A is G-invariant.

(2.4) Assume the hypothesis and notation of2.3 with K = CG(A) ~A and.# nonemply. Then .# ~ COU!(A)(OutG(A».

Proof As.# is nonempty, we saw in the proof of 2.3 that there exists anisomorphism rp: K ----. A such that G = AM, where M is the stabilizer of rp inG. Let g ----. ug and g ----. fJg be the conjugation maps of G into Aut(A) andAut(K), respectively. Hom*(K,A) = {u ° rp: u E Aut(A)} and for'1/ E Hom *(K, A), g E G, and k E K, (g'l/)(k) = g('I/(k g» = (ug° '1/ ° fJ; ')(k),so g'l/ = ug° '1/ ° fJ; I. As fJg = 1 for g E A, 'l/A = Inn(A )'1/, so n: 'l/A ----. Inn(A)'I/is a bijection of ,Jrom*(K, A) with Out(A) = Aut(A )/Inn(A). Also forgEM, grp = rp, so rp ° fJ;' = u;' 0 rp, and thus writing '1/ = U 0 rp, g'l/ = ug °'1/ ° fJ;' = Cag)u) orp, so n defines an M-isomorphism, where M acts onOut(A) by conjugation. As A is in the kernel of both actions, n is also a Gisomorphism so .# is isomorphic to the fixed point set COU!(A)(OutG(A» ofG acting on Out(A) by conjugation.


3. THE PROOF OF THEOREM 2

55

In this section, actions of groups on sets will be on the left. Given groupelements x and y we write "y for yX- 1 = xyx - I. If S is a group acting on agroup N, the semidirect product will be denoted by N.S. If S acts on a set nand L is a group, the wreath product L wr S = L wr fl S is the semidirectproduct D.S where D = Il fl L consists of all functions f: n ~ Land S actson D via

Cf)(a S) = f(a S

), a E n, s E S

where by definition as = s~ I.

When n is the set S/V of left cosets tV of a subgroup V of S, it will oftenbe convenient to view the functions in Iln L as defined on S, but constanton the cosets of V. That is, we let S act on itself by left multiplication andembed L wr fl S in L wrs S via the injection

L wr fl S ~ L wrs S

fs f---->f;

where f': S ~ L is defined by f' (tu) = f(tU) for t E Sand u E V. Subject tothis identification, Eq. (*) becomes

('f)(t) = f(s -I t), s, tE S.

The group L wrs S has a useful faitful permutation representation on theset L X S given by

!S(x, t) = (f(st)x, st), fE IlsL, xEL, s, tE S.

The embedding of L wr fl S in L wrs S then induces a faithful action of theformer group on L X S.

(3.1) Generalized Wreath Products

In the remainder of this section X denotes a group with a normal subgroupS = IlaEfl La' which is a direct product of groups La isomorphic to Landpermuted transitively according to an action of X on the index set n; thus

xEX, aEn.

Moreover except in 3.5 we assume X = D.S is the semidirect product of Dand some subgroup S.

There are two useful ways to identify D with functions f on n. Thesimplest is to letf(a) take values in La' and identify fwith IlaEflf(a) E D.The action of S on D then translates into

Cf)(a) = S(f(S-l a )), sE S, aEn.


A second point of view, used in our earlier discussion of the wreath product,is to use the isomorphism of L with La and letftake values in L; this gives

sE S, aEn

for some automorphism rs(a) of L. In the case of wreath product, rs(a) = 1for each s E S and a E n, as reflected in Eq. (*), after suitable translation ofnotation. More generally we have an embedding of X in (L. Aut(L)) wr 0 Sextending the natural inclusion no L -> no (L. Aut(L)), and mapping s E Sto srs' where rs: n -> Aut(L) is the function mapping a to rs(a). Havingpointed out this embedding, in the remainder of this section we adopt thesecond point of view only for wreath products. In both cases we use theconvention (introduced earlier) that functions on n may be viewed asfunctions on S constant on the cosets of the stabilizer U in S of some pointof n.

(3.2) The Correspondence Going Down

Again U is the stabilizer in S of a point of n, so we regard n as the cosetspace S/U. Let D' = naEO-1U1 La' so that U acts on D' and DU/D' ~ L.U.Let n: DU -> DU/D' be the natural map. If S' is a complement to D in Xthen U' = S' n DU is a complement to D in DU, so n(U') is a complementto neD) ~ L in n(DU) ~ L.U. Let Ii(S') be the image of n(U') in L.U. If Sffis a complement conjugate to S' in X then Sff is conjugate to S' under D, soIi(Sff) is conjugate to Ii(S') under L. We now establish the converse.

(3.3) Let S' and Sff be complements to D in X, and suppose Ii(S') isconjugate to Ii(Sff) in L.U. Then S' is conjugate to Sff in X. Indeed ifIi(S') = Ii(Sff) then S' is conjugate to Sff under the action of any subgroupDo of D of the form no L o with L o~ L and DoS' = DOSff.

Proof Without loss we may take Ii(S')=Ii(Sff). Let ({J:X->(L.Aut(L)) wr0 S be the embedding described in 3.1. By definition of ({J, ({J(S) isa complement to D* = no (L. Aut(L)) in D*.S, and then as S' and Sff arecomplements to D in D.S. and ({J(D) = ({J(DS)n D*, ({J(S') and ((J(Sff) arecomplements to D* in D*S. Ii(S') = Ii(Sff) so as ({J(D) = ((J(DS) n D*, wehave n*(({J(U')) = n*(({J(Uff )) and then Ii *(({J(Sff)) = 1i*(({J(S")), using theobvious notation. Now if L o and Do are as in the statement of the lemma,and ({J(S') is conjugate to ({J(Sff) under ({J(Do)' the certainly S' is conjugate toSft under Do. Hence, passing to D*S and preforming a suitable change ofnotation, we may assume X = L wr 0 S is a wreath product.

For s E S, Ds n S' and Ds n Sft contain unique elements which wedenote by s' and Sff and let fJ s = s' S -I and Ys = S ffS - 1 be the correspondingelements of D. Notice that s'=fJsy;lsff, so as DoS'=DoS ff , fJsy;lEDo'


Regarding fl s and Ys as functions from n into L, this is equivalent to theassertion that fls(t) ys(t) -I E L o for each t E S.

Since /1(S') = /1(S") we have n(U') = n(U"), where U' = s' n DU andU"=S"nDU. Thus for uEU, n(u')=n(u"), so fluy;;IEker(n)=D u ;equivalently flu(U) = yu(U), or using our convention embedding X inL wrs S, this becomes flu(l) = yu(l). Now from (**) we obtainsflJs) = sYu(s) for all s E S. As a consequence of the definition of fl s and Yswe have fl su = fl ssflu and Ysu = Ys sYu' Then recalling that fl su and Ysu areconstant on the cosets of U, we have flsu(su) ysu(SU)-1 =flsu(s) ysu(S)-1 =fls(s) y,(s) -I. Hence if we define 15 E Os L by b(s) = fls(s) ys(S)-1 we have 15constant on the cosets of U, so 15 E OQ L = D. Indeed as fls(s) ys(S)-1 E Lo,even 15 E Do.

Define 15' and 15" in OsL by b'(s)=fl,(s) and b"(s)=ys(s). To show 15conjugates S" to S' it is enough to show 15' conjugates S to S' and 15"conjugates S to S". For this we use the faithful action of L wrs Son L X S.First note that by (3*)

h'S(X, 1) = (15' (s)x, s) = (fJs(s)x, s) = 13 sS(x, 1) = S'(x, 1)

for all s E S and x E L. So for s E S

h'S(X, t) = h'SI(X, 1) = (st)'(x, 1) = S'I'(X, 1)

= S'h'I(X, 1)= S'h'(X,t).

We have shown b's = s'b', so that 15' conjugates S to S' as desired. Thesame argument shows 15" conjugates S to S", completing the proof of thelemma.

(3.4) The Correspondence Going Up

Recall the discussion of I-cohomology in Section 2. By the hypothesis ofthis section we have an action of S on D and an action of U on L via theidentification of L with the factor La of D stabilized by U. So T(S, D) andT(U, L) are defined.

Let:}" be a set of right coset representatives for U in S containing 1 as therepresentative of the coset U. Define u: S -+ U by

Y E:}", s E S nyu.

For x, s E Sand y E nU, L) define

and define

58

Observe first that

ASCHBACHER AND SCOTT

u(vs) = VU(s) for all s E S, v E U

and hence by the cocycle condition we have

Yu(VS) = Y~Yu(S)'

Then for v E U and x E X,

i.(vx) = (Y;(~X)Yu("XS))vx= «Y~Yu(X)) -I Y~YU(Xs))"x

= (y;(~) YU(XS)Y =i'(x).

Hence for each s E S, is: x -4 is(x) is a function from S into L constant onthe right cosets of U, so ys E nQ L :(: ns L. Thus y: s H ys is a functionfrom S into nQ L = D.

We show next that y satisfies the cocycle condition. For if s, t, xES, thenby (4*) we have

and then applying the definitions of y, is' and!r, we get

ys(x)' (Y/(S-IX)) =i.(x-I)S (!r(x-1s))

(-I )X- 1S« --I )X--1S)= YU(X-lj YU(X-1s) YU(X-'s) YU(x-lSI)

= (Y;(_~_I) YU(X-ISl)y-1 =h/(x- I) = Ys/(x).

So the cocycle condition holds and hence

Y H Y is a map from T(U, L) into T(S, D).

Next y*(S) consists of the elements Yss, s E S, so ,u(y*(S)) consists of theelements yJl)v, v E U. As Yv(l) =iv(1) = Y;(~)YU(V) = Y", we conclude:

,u(y*(S)) = y*(u). (h)

(3.5) The Nonsplit Case

Together with (3.3) the result (7*) establishes Theorem 2 below under thehypothesis that X is a semidirect product D.S. We now drop the assumptionthat D has a complement in X and show

D has a complement in X if n(D) has a complement in n(Nx(L)). (8*)

Suppose WID' is a complement to DID' in NG(L)ID', and consider thehomomorphism X -4 Sym(Xlw), where the latter is the symmetric group on


the set of right cosets of W in X. The kernel is contained in Wand intersectsD trivially, so without loss we may assume the kernel is I and the map is aninclusion. Note L acts regularly on the points in L W/W, but fixes all therest. No point is fixed by D, and the points moved by La' L IJ are disjoint fordistinct a, {3.

The normalizer N of D in Sym(X/W) can now be seen to have a very niceform. The normalizer of L induces on the support of L a group £ in which Lis a regular normal subgroup, and we have

N = (Q La) Sym(Q) ~ Lwro Sym(Q).

Since L is regular in £, it has a complement, call it V. Then (no Va)Sym(Q) ~ V wr 0 Sym(Q) is a complement to D in N. It follows that D has acomplement in X as well, and (8*) is proved.

With a little further study of N above we could even reprove (7*).However, we will need the cocycles again momentarily. Meanwhile observethat (3.3), (7*), and (8*) establish Theorem 2.

The following section is not needed elsewhere in this paper.

(3.6) Stability

Notice that a complement V to D in NAL) is not necessarily contained ina complement T to D in X; we are only guaranteed D' V = D'NT(L). Whatconditions on V guarantee V ~ T, or, equivalently, V = N AL) for somecomplement T to D in X?

PROPOSITION. If V is a complement to D in NAL), then V~ T for somecomplement T to D in X if and only if

D V x n V is D-conjugate to V x n D V for each x E X.

Proof First suppose V ~ T. Note that the condition on x depends onlyon the D-coset to which it belongs. Hence we may take x E T. But now bothV and V* are contained in T, and we have

D V X n V = D V X n Tn V = v x n V = V' n Tn D V = v x n D V.

Next suppose the conjugacy condition is satisfied. We will use the methodof (3.4), carefully choosing the set j" of right coset representatives for U inS required there. Let .z be a set of (U, U) double coset representatives in S,and for each z E .z, let J"z be a set of right coset representatives for UZ n Uin U. Finally, put j" = UZE$zj"z'

Let {3E F(U,D) and yEF(U,L) be cocycles corresponding to V. Thus{3,(l)=r, for sEU. Choose dzED with (DVznV)d,=vznDV. Let


s E un zu and t = sZ. Then t E U ~ DV, so (fJss)' E Vz nDV. AlsofJttEDVznV, so (fJtt)d'E VznDV. As D(fJtt)d'=Dt=D(fJss)' weconclude (fJss)' = (fJ,t)d,. Now 'as t acts on L Z we conclude (fJ,(Z-I)t)d, =fJ~(z -I)t = (fJs( I»Z . t (by (4*» =Y~ . t, which we record:

for each s E un zU and t = sZ.

Recall the definition of u(x) and y from (3.4). As z E:P', u(z) = 1, soYU(z) = 1. Also zt = sz with s E U, so u(zt) = s. Then y,(z -I) =(Y;(~) YU(zl))' = Y~, so from (9*) we have

for each t E un U z•

Let

D~= fl M,MEr-luI

fJ~ and y~ the projections of fJu and Yu on Dr- and fJ', l' the corresponding maps of U into Dr' Then fJ', l' E F( U, Dr) and by (10*),(fJ')*(U n UZ)d, = (y')*(U n UZ) mod D~, so by (3.3) there exists ez E Drwith (fJ')*(U)d: = (1')*(U). Let e = nZE.J"eZ' Then Ve = fJ*(UY = y*(U) =y*(S) n UD, completing the proof.

We conclude this section by remarking that Theorem 2 may be regardedas a strong version for nonabe1ian I-cohomology of Shapiro's lemma inabelian cohomology. Similarly the proposition above is a generalization of a(less wel1 known) stability property of the cohomology of induced modules.


Theorem 3 was first obtained in col1aboration with R. Guralnick usingelementary group theoretic techniques. We give a cohomological proof here.We first recall some facts about cohomology; standard references are [21and [31. If U is a module for a group Q, then H°(Q, U) denotes the fixedpoints of Q on U, as usual.

Assume throughout this section that G is a finite group, p is a prime, K isa field of characteristic p, and V is an irreducible KG-module. Observe thatas V is irreducible we have:

(4.1) If Q is a normal subgroup of G with rv, Q1* 0, thenHO(Q, V) = O. In particular if G is faithful on V and I * Q <J G, thenV= [Q, V1 so H°(Q, V)=O.

We need the following lemma which can be derived from 4.1 and theHochschild-Serre sequence [3, (10.6)] or [5, p. 126, Remarque1:


(4.2) H1(G, V) ~ HO(G/Q, HI(Q, V)) for each normal subgroup Q of Gwith [Q, V] *' O. In particular if H1(G, V) *' 0 then H1(Q, V) *' O.

Here we are using the fact that as Q <l G there is a natural action of G onHI (Q, V) with Q in the kernel of this action. We also need the followingresult:

(4.3) Let Sand T be irreducible KA and KB-modules, respectively, Then

Finally, we prove the following elementary result:

(4.4) Let X <l G with G/X solvable and V = IV, Xl. Thendim(H1(G, V)) ~ dim(H1(X, E)) for each irreducible KX-submodule E of V.

Choose a counterexample to 4.4 with G/X of minimal order, letX ~ Y <l G with Y maximal in G, and let A be an irreducible KY-submoduleof V. If Y *' X then by minimality of G/X, dim(H1(G, V)) ~ dim(H1(y, A) ~dim(H I (X, E)) for each irreducible KX-submodule E of A. As all irreduciblesubmodules for X on V are isomorphic to some G-conjugate of E, the lemmaholds, contrary to our choice of G as a counterexample. Thus X is maximalin G, so G = (g, X) for some g E G.

Form the semidirect product XV and let U= CAU!(VX)(V)n ceXV/V). Werecall U is an elementary abelian p-group with U/V ~ H1(X, V), and thatthe action of G on V induces an action on U, and hence also on U/ V, withthe latter action equivalent to the natural action of G on H1(X, V). Inparticular as V = (EG), H1(X, V) = U/V = [J = (15G), where D/E = CU/E(X),and 15 ~ HI (X, E). Choose n maximal subject to Vo = (pi: 0 ~ i ~ n) =

c±) 7~o Eg'. As X is irreducible on E and X <l G, pi ~ Vo for all i, so V = Voas V=(EG). So [J=c±)7=o15g

i. H1(G, V)=HO(g, D), so we must showdimeCv(g)) ~ dim(15). Assume otherwise and let W= (15gi: 0 ~ i < n). Thenthere is 0 *' wE Cw(g). Let Wi be the projection of W on 15gi. Then L Wi =w=wg=Lwf with wfE15gi+', so Wf=W i + 1 for 0~i<n-1, andw~ I = wo' Therefore W~_l = Wo E 15gn n 15 = 0, so by induction on i,}i"'i+ I = wJ = 0 for 0 ~ i < n - 1. But now w= 0, a contradiction. Thus 4.4 isestablished.

Now to the proof of Theorem 3. Assume that G is faithful on V andH1(G, V) *' O. If k is an extension of K then H1(G, V 0 k) ~ H1(G, V) 0 k,so replacing K by an algebraically closed extension, we may assume K isalgebraically closed. By 4.1, Op(G) = 1. H1(Op,(G), V) = 0 (by a Frattiniargument for example) so by 4.2, 0 p ,(G) = 1. Hence M = F* (G) is the directproduct of simple components. Let L be one of these components andX = CM(L). By Clifford's theorem, V is a semisimple KM-module whose

481/92/15


homogeneous components Vi' 1~ i ~ n, are permuted transitively by G. By4.2, H1(M, V) *- 0, so HI(M, Vi) *- 0 for some i, and then by the transitiveaction of G, for all i. Similarly HI(M, W) *- 0 for each irreducible KMsubmodule W of V.

As M = L X X and K is algebraically closed, W ~ S ® T for irreducibleKL and KX submodules Sand T of V, respectively. Choose W so thatfL, W1 *- O. Then S = IL, S1, so HO(L, S) = O. Hence by (4.3),o*- HO (X, T), and then as T is a simple KX-module, [T, Xl = o. We haveshown that W = IL, W] and IX, Wj = 0, so Vi = IVi' L] and IX, Vi] = 0 forany homogeneous component Vi with IVi' L ] *- O. As G is transitive on thehomogeneous components of M, it follows that (1) and (2) hold. Now byShapiro's lemma 11 (or Theorem 2):

(4.5) HI(G, V) ~ HI (NG(L), U).

Let Y = CG(L), A = NG(L), and B = YL. As G is irreducible on V, A IS

irreducible on U. From (4.2) we get:

(4.6) HI(A, U) ~ RO(A/B, RI(B, U)).

Then from 4.5 and 4.6 we conclude HI(B, U) *- O. Next applying theargument of the last paragraph to the action of B = Y X L on U we obtain(3). After that we observe:

For if not there is a complement B' to U in BU with Y*-B' Ii YU= Y'. By(3), U~ Z(YU), while Y' <J B' so Y' <J B'U=BU. Thus as IL, YU] = U,IL, Y'] ~ Y'li U = 0, so Y' = Cyu(L) = Y, a contradiction. So 4.7 isestablished.

From 4.5-4.7 we get HI(G, V) ~ HO(A/B, HI(L, U)) ~ HO(A/Y/B/Y,RI(B/Y, U))~HO(A/Y,U), with the last isomorphism following from 4.2.Therefore (4) holds.

Finally, from the classification of the finite simple groups, the Schreierconjecture holds, so Out(L) is solvable. Hence (4) and 4.4 imply (5).

This completes the proof of Theorem 4.The Schreier conjecture, and hence the classification of the finite simple

groups, is used in the proof of part (5) of Theorem 3, and, via 6.3 and 6.5, toshow the map in part (C)(3) of Theorem 1 is a surjection. These results arenot needed elsewhere in the paper, which is otherwise independent of theclassification.


5. MORE ON DIAGONAL SUBGROUPS

63

In this section we continue the hypothesis and notation of Section 2. Thefixed points of the action of G on A will be denoted by CA (G) or HO (G, A).The sets ~(G,A) and H1(G,A) are "pointed" in that they havedistinguished elements: The identity of the group HO( G, A) and the trivialclass in H1(G,A).

If a: X ---> Y and fl: Y ---> Z are maps of pointed sets with distinguishedpoints Xo, Yo, and Zo in X, Y, and Z, respectively, then

is said to be exact if the image of X under a is the full preimage in Y of Zo

under fl.Suppose now G acts on a group B and A is a G-stable subgroup of B. Let

C be the coset space B/A = {bA: b E B}, and notice that CB(G) actsnaturally by left multiplication on HO(G, C), with the latter regarded as apointed set with distinguished point the coset A.

(5.1) There is a natural exact sequence of pointed sets

in which the map J is given by J(bA) = [y] where Yg= b- 1 gb, for g E G, andthe remaining maps are the obvious ones. Moreover the fibers of J are theorbits of CB(G) in its natural action by left multiplication on HO(G, C).

Proof Except for the last assertion and the fact that we have notassumed B normal in A, this is just Proposition 1 on page 133 of [5]. Thedetails are straightforward with the possible exception of the transitivity ofCB(G) on the fibers of J. Suppose J(bA) = J(dA) for some bA, dA E C. Thenthere is aEA with ab-l(gb)ga~I)=d-l(gd) for all gEG, so thatx = dab-I E CB(G). But of course xbA = dA.

Apparently the first person to write down this sequence without thenormalizer assumption was Giraud in his book "Cohomologie nonabeliene."

In the remainder of this section we assume:

HYPOTHESIS 5.2. G is a group containing a normal subgroup D which isthe direct product of the G-conjugates (L/: i E I) of some subgroup L. Let f0be the set of D-classes of full diagonal subgroups of D (as defined inSection 1) and let .f0(G) consist of the classes in .f0 stable under Gconjugation.


As we saw in the introduction, @(G) is important in the study of maximalsubgroups. Our aim here is to give a parameterization of @(G) when @(G)is nonempty; this is accomplished in Theorem 5 below. We do not addressthe question of whether @(G) is nonempty. Thus the situation is analgous toa familiar one in the theory of group extensions, where one asks first if theextension splits, and if the answer is positive one has at least a theoreticalparameterization of the classes of complements by HI, which may becomputed explicitly in favorable circumstances.

To begin, let Auto(D) denote the subgroup of Aut(D) fixing L; for eachi E I. Then Auto(D) is the direct product of the groups Aut(L j ), i E I. LetOuto(D) be the image of Auto(D) in Out(D) and assume Af E@(G). Thisimplies G=DNdAI)' By 1.2, Auto(D) is transitive on the full diagonalsubgroups of D, and by 1.1 the stabilizer in Auto(D) of A 1 is a full diagonalsubgroup isomorphic to Aut(A I); we identify Aut(A I) with this stabilizer.The action of Auto(D) induces a transitive action of Outo(D) on @, in whichthe stabilizer of A f is Out(A I)' so we have a bijection of sets @ ;::::: Outo(D)/Out(A I). We also have the conjugation map G -. Aut(D), whose imageAutG(D) acts on Aut(A I) since G = NG(A I)D, and also acts on the normalsubgroup Auto(D) of Aut(D). This induces the map G -. Out(D) whoseimage OutG(D) acts on Out(A I) and Outo(D). The permutation representation of G on @ factors through the map G -. OutdD) andOutG(D) Out(A I) is the stabilizer of A f and the coset Out(A I) in the transitive representations of OutG(D) Outo(D) on @ and Outo(D)/Out(A I),respectively, so the isomorphism @;:::::Outo(D)/Out(A 1) is G-equivariant. Werecord these observations as:

(5.3) There is an isomorphism of sets @ ;::::: Outo(D)/Out(A I) defined by(IlA 1t-..ffOut(A I), where IJEAuto(D) and.ff is the image oflJ in Outo(D).This isomorphism is G-equivariant.

From 5.1 and 5.3 we obtain:

(5.4) There is an exact sequence of pointed sets

1 -. HO(G, Out(A I» -. HO(G, Outo(D» -. @(G)~ HI(G, Out(A I»

-. H1(G, Outo(D»

with Af the distinguished point of @(G) and with () and its fibers describedin 5.1.

We next observe that by 1.1.3:

(5.5) G acts by inner automorphisms on Out(A I), in the sense of 2.2.Indeed OutG(D) Out(A I) = Out(A I) X Y(A I)' where Y(A I) is the group ofouter automorphisms induced on D by CAut(A I) AutG(D)(A I)'


Hence we can apply 2.2 to conclude:

65

(5.6) HI(G, Out(A I)) is isomorphic to ,¥om(G,Out(A I)) under theisomorphism of 2.2. This isomorphism together with the projection of Out(A I)onto Out(L) induces an isomorphism H\G, Out(A I)) ~ ,crom(G, Out(L)).

Let AD E @(G) and define VA: G -t Out(A) to be the composition of themap G -t OutG(D) with the projection Out(A) OutG(D) -t Out(A) supplied by5.5. Further let v~: G -t Out(L) be the composition of VA with the projectionOut(A)-t Out(L).

(5.7) v1 depends only on the class AD, not on the representative A.

Proof Out(A) = Out(A X) for each xED since Aut(A) Inn(D) = Aut(A X)Inn(D).

We can now state the main result of this section.

THEOREM 5. Let G satisfy Hypothesis 5.2 and assume @(G) isnonempty. Let Homo(G, Out(L)) denote the set of homomorphisms inHom(G,Out(L)) whose restriction to NG(L) is the conjugation mapNG(L) -t Out(L). Then the map

v: @(G)-t Homo(G, Out(L))

ADf------>Jfiis a bijection.

The proof of Theorem 5 involves a series of lemmas.

(5.8) v~ E Homo(G, Out(L)).

Proof Let M = NG(L). The conjugation map M -t Out(L) is thecomposition of M -t Out(D) with the projection 'TrL:OutN(L)(D) -t Out(L). Onthe other hand V~ 1M is the composition of M -t Out(D) with the projections'TrA : OutM(D) Out(A) -t Out(A) and 'TrL, so it suffices to show 'TrL 0 'TrA = 'TrL. AsAut(A) AutM(D) commutes with the projection A -t L, ker(nA ) -< ker('TrL).Also 'TrT,: Out(A) -t Out(L) is an isomorphism, so indeed 'TrL 0 7l:A = 7l:L.

Tracing through the definitions and using 5.1, 5.3, 5.4, and 5.6, we conclude:

(5.9) (1) I5(A D) E HI(G, Out(A I)) is the cohomology class of thecocycle gf------>iJ-1(g{J), where (JEAutoCD) with /}A1=A and iJ denotes theimage of (J in Outo(D).

(2) The image of I5(A D) in o?om(G, Out(A-.J) under the isomorphismof 2.2 is the orbit of the homomorphism g f------> iJ- \g{J) i(g), where i = VA ': G-tOut(A I)'

66 ASCHBACHER AND scon

(5.10) (1) ThereexistsfJEAuto(D)nC(L) with IlA I =A.

(2) If fJ is chosen as in part (1), then the composition of thehomomorphism g f-----+ iJ-I(gfJ) i(g) with the projection Out(A I) ---> Out(L) isjust the map v1.

Proof As Auto(D) = Aut(A I )(Auto(D) n C(L)) part (l) holds. Choose fJas in part (1). G = NG(A I)D and D is in the kernel of both the map n: g--->iJ- I(gfJ) i(g) and v1, so it suffices to show v1 and the composition of n withthe projection Out(A I) ---> Ou!Q-) agree on NG(A I)'

Let g E NG(A I) and iJ-I(gfJ) = a, so that a E Out(A I) and n(g) = (JAg)·Let ~ be the image of g in Out(D). Then gfJ = tiJ, so ~ = iJ . iJ-I (gfJ) t,fJ - 1 =iJat,fJ-l = ll"(a~). Now a~ acts on Out(A I) and IlA I = A, so ~ = ll"(a~) acts onOut(A). Thus ~=~1~2 with ~IEOut(A) and ~2E Y(A) by 5.5. Notice thatv1(g) is the projection of ~I on Out(L). Next ~ll" = ~r~[ with ~r E Out(A I)and ~[E Y(A I)' and as the projection of iJ on Out(L) is trivial, ~1 and ~rhave the same projection v1 (g) on Out(L). Finally, ~ll" = a~ witha E Out(A I)' so ai(g) is the projection of ~ll" on Out(A I); that is, n(g) =ai(g) = ~r. Therefore v1 (g) = projection of ~r on Out(L) = projection ofn(g) on Out(L), completing the proof.

(5.11) Let romo(G,Out(L)) denote the image of Homo(G,Out(L))under the natural map assigning a homomorphism to its Out(L )-conjugacyclass. Then romo(G,Out(L)) is the image of t5(0J(G)) under theisomorphism HI(G, Out(A I)) ~rom(G, Out(L)) of 5.6.

Proof We have the following commutative diagram:

HI(G, Out(A I)) ~ HI(G,Outo(D))

b1 1cHI(NG(L), Out(A 1))~ HI (NG(L), Out(L))

The map a is induced by the inclusion Out(A I) ---> Outo(D), and by 5.4,t:5(@(G)) = a-I(zo), where Zo is the distinguished point of H1(G, Outo(D)).The map d is the isomorphism induced by the projection Out(A I) ---> Out(L).The map b is the restriction map taking [y] E HI(G, Out(A 1)) to [yIN(L)] EHI (N(L), Out(A I )). Finally, the map c takes [y] E H 1(G, Outo(D)) to[n 0 yIN(L)]' where n: Outo(D)---> Out(L) is the projection. By Theorem 2, c isan isomorphism. As c and d are isomorphisms, b-l(xO) = a -I(ZO) = t:5(01(G)),where xo is the distinguished point of HI (N(L), Out(A I))'


We have a second commutative diagram:

H1(G, Out(A 1)) ~ orom(G,Out(A 1))

b1 1BHI (NdL ), Out(A 1))~ ,~om(NdL), Out(A 1))

67

Here e is the isomorphism of 5.6, f is the restriction of e toH1(NG(L),Out(A 1)), and B is defined by B([qJ]) = [qJIN(L)]' By definition offin 2.2,J(xo) is the Out(A 1)-orbit of VA '. Hence

so that the lemma holds.Let us review our progress to this point. We have the diagram:

g(G) ~ b(g(G))

-1 ,1Homo(G, Out(L))~ oromo(G,Out(L))

where b is the map of 5.3, v: AD H v1, r is the restriction of the map of 5.6to b(g (G)), and s is the natural map assigning a homomorphism to itsOut(L )-orbit. By definition band s are surjections, and by 5.11, r is anisomorphism. By 5.9 and 5.10, the diagram commutes.

To prove Theorem 5 we must show v is an isomorphism, and from the lastparagraph this is equivalent to the assertion that v induces isomorphismsv: b -I (x) H S -I (r(x)) on the fibers of band s. The projection Couto(D)(G)->Cout(L)(NG(L)) defines an isomorphism which, by 5.1 and 5.4, induces atransitive action of Cout(L)(NG(L)) = M on the fibers of b. By definition of s,M acts on the fibers of s. Thus to establish Theorem 5 it suffices to prove thefollowing three lemmas:

(5.12) M acts transitively on the fibers of s.

(5.13) v commutes with the actions of M on the fibers of band s.

(5.14) Cout(L)(vi(G)) is the stabilizer in M of both vi and AD.

We first prove 5.12. Suppose qJ and If/ are in the same fiber of s. Thenxlf/ = qJ for some x E Out(L). As qJ, If/ E Homo(G, Out(L)), If/(g) = qJ(g) =i(g), for g E NG(L), where i(g) is the image of g in Out(L) under the


conjugation map. Thus Xi(g) = i(g) for each g E NG(L), so x E M,establishing 5.12.

We next prove 5.14. First consider the stabilizer M i in M of i = lit. x E Mfixes i precisely when i(g) = i(gr for all g E G. Thus M i = CM(i(G)). AsM = COUI(Lj(i(NdL)), M i = COUI(Lj(i(G)) as claimed. Next consider thestabilizer N of A v in M. By 5.3 and the definition of the action of M on @,

N is the image under the projection map of the stabilizer No in COUI (Dj(G) ofthe coset Out(A) E Outo(D)/Out(A). But No = COUI(A)(G) = COUI(AOj(VA(G)),so N = COUI(L)(i(G)), and 5.14 is established.

Finally, we prove 5.13. For x E Auto(D) let x denote the image of x inOuto(D). Let ~ E M, ~ a preimage of ~ in Aut(L), and f5 E Auto(D)such that ~ is the projection of f5 and g= is the preimage of ~ under theprojection isomorphism COUlo(D)(G) ...... M. Further let a be the preimage of~"

in Aut(A) under the projection isomorphism Aut(A) ...... Aut(L). ThenfJ=f5a- 1 EAuto(D)nC(L) with ~A=IlA, and by 5.10, V«~A)D) is thecomposition of the map g f------> iJ-I (KfJ) i(g) with the projection isomorphismp:Out(A) ...... Out(L), where i=vA

• Now iJ-I(KfJ)i(g)=af5-1(Kf5Wa-l)i(g)=iWa-l)i(g), as I'fECOUlo(D)(G). Moreover aEOut(A), so (Ka )=i(g)ai(g)-', and thus a(Ka-l)i(g)=ai(K)' Hence V«~A)D)=poiii=

(~L)(vD. However, (~)D is the image of AD under g;" EM, so 5.13 isestablished.

This completes the proof of Theorem 5.We close this section with a corollary of Theorem 5.

(5.15) Assume Hypothesis 5.2 and assume also that G = (NG(L), x) forsome x E G. Then 19(G)1 ~ IOut(L)I.

Proof Each qJ E Homo(G, Out(L)) is determined by qJ(x), so Theorem 5gives the indicated bound.


In this section G is a finite group and MEn with kerM(G) = 1; that isME n*. Let Ooo(G) denote the largest solvable normal subgroup of G. IfOoo(G) oF 1 a well-known elementary argument shows that F*(G) is anelementary abelian p-group, M is a complement to F*(G) in G, and M actsirreducibly on F*(G) via conjugation. Indeed ~* ~ H1(M, F*(G)), andhence part (A) of Theorem 1 holds by Theorem 3 (established in Section 4).

So we may assume throughout the remainder of this section that0oo(G) = 1. Hence F*(G) is the direct product of the components of G, andeach component is simple. Let L be a component of G, A = L G, andD = (A). This notation differs slightly from that in the statement of


Theorem 1 in the introduction where F*(G) = D, but in the next lemma wedeal with case (B), hence quickly reducing to the case F*(G) = D. As in theintroduction, ~ denotes the conjugacy class of maximal subgroups of G and~* the set of classes with a representative in n*. We also use the othernotation and terminology established in the introduction.

(6.1) As.sume F*(G)=I=-D. Then

(1) There exists a component K of G and an isomorphism a: K ---> Lsuch that K E,1, NdL) = NG(K), F*(G) = «LK)G), CdD) = (KG) ~ D,LK (I M = diag(a), and F*(G) (I M is the direct product of the M-conjugatesof diag(a).

(2) Let ~(L) consist of those U in KG with NdL) = NdU) andS?w(NG(L)) =I=- 0, and let Y denote UUE ~(L) 2?1w (Nd UL)). Then forNEn* there exists U(N)Et/(L) such that the map rp:NGf------>(N (I LU(N))W(N) is a bijection between ~* and Y.

(3) '&'* ~ UUEi'(L) Cout(L)(NG(L)).

Proof Let 1 =I=- A <J G with A ~ CG(D). Set B = AD (I M andAM = (A (I M)A). AD = AD (I G = AD (I MD = BD. Then AM =«A (I M)A) = «(A (I M)AD) = «A (I M)BD) = «(A (IMt) ~ M. Also AM isinvariant under M and D, so AM <J MD = G. Hence as kerM(G) = 1, AM = I,so A (I M = 1. Therefore CM(D) = 1.

We can interchange the roles of A and D in the argument of the lastparagraph to show AD = AB and D (I M = 1. Then A ~ ADID = BDID ~ Band similarly D ~ B. As this holds both for A = CG(D) and for a minimalnormal subgroup A of G contained in CG(D), we conclude A = CG(D) is aminimal normal subgroup of G isomorphic to D. In particular F*(G) = AD.

We have also shown that B = AD (I M = diag(fi) is a full diagonalsubgroup of AD, where (3: D ---> A is the composition of the isomorphismsni;J: D ---> Band 7CA : B ---> A, and TCD and TCA are the projections of Bon D andA, respectively. As these porjections commute with the action of M byconjugation, so does (3. Thus setting K = L III NM(K) = NM(L), soNG(K) = NG(L). Let a = (3IL' Then diag(a) = B (ILK = M nLK, and (I) isestablished.

Next D ~NG(LK), so NG(LK) = DNM(LK) and hence (MnLK)LK =(M n LK)DK is invariant under NdLK). Adopt the notation of 6.1.2. ThenK = U(M) E ~ (L) and rp takes '&' * into !/.

Next let KE ~(L), XLK E cY', and n the set of subgroups OWE(LK)GXW,X wE X G n W. By (1.3, D is transitive on n so by a Frattini argument,G = DNG(y), for YE n. By 1.1 and 1.2, X = NLK(X), so Y = ND(y). AsG=DNdY), NG(y) is maximal in G by 1.5. Hence rp is a surjection.Suppose X = LK nM. Then by (1), MnD E n, so M=NG(MnD) ENG(Y)G. Thus rp is an injection and (2) is established.


Part (3) follows from 2.4 and (2).Notice that Lemma 6.1 implies that the conclusions of part (B) of

Theorem 1 hold when F*(G)*- (L G). Thus in the remainder of this section

we assume that F*(G) = (L G) = (.1) = D.

(6.2) Either

(1) AutM(L) = AutG(L) and L n M = 1, or

(2) AutM(L) is a maximal subgroup of AutG(L), AutM(L) n L = M nL *- L, and M n D is the direct product of the M-conjugates of M n L.

Proof Let X=NG(L) and X*=X/CG(L)=AutdL). D<.NG(L), soNG(L) = NM(L)D and then X* = L*NM(L)*.

Suppose L * <. NM(L) *. Then X* = NM(L) *; that is, AutG(L) = AutM(L).So L<.NM(L)CG(L). Now if I*-LnM then L=«(LnM)L)=«(L n M)LCG(L» <. «(L n M)NMiLlCG(L» = «(L n M)N,rf,L» <. M, contradictingkerM(G) = 1. Thus L n M = 1 and (1) holds.

So assume L * 4;; NM(L) *. Let N* E "lX' contain NM(L) *. As X* =L*NM(L)* and NM(L)* <,N*, we have N* = (L nN)* NM(L)*. LetB = «(L n N)M). As L n N <J NM(L), B is the direct product of the Mconjugates of L n N. Then MB <. G with BNM(L ) = NMB(L ), soNMB(L)* = N*. Hence MB is a proper subgroup of G, so as M is maximal,B <. MB = M. If d E M n D - B then as M is transitive on .1 and B <J M,we may choose d so that d* EB* = (L nN)*. Then as N* E nx"X* = (d*, N*) <. NM(L )*, contrary to our assumption. Thus M n D = Band (2) holds.

We now partition n* into three subsets according to the three subcases ofcase (C) of Theorem 1:

n{ = {N E n*: AutN(L) = AutG(L) and NnD = I}

n! = {NE n*: AutN(L) = AutG(L) and N nD *- I}

nt = {N E "l *: AutN(L) is maximal in AutG(L)}.

Let 'Tt';* denote those classes in 'Tt' * with a representative in n r We record alemma restricting members of n t.

(6.3) AssumeD=L. ThenLnM*-1 andM=NdLnM).

Proof L n M <J M, so if L n M *- 1 then M = NG(L n M) bymaximality of M. So assume L n M = 1. Then G is the semidirect product ofL with M. Then M is isomorphic to a subgroup of Out(L), so M is solvable.Let X be a minimal normal subgroup of M. Then X is a p-group for someprime p. By maximality of M, M = NdX). So CL(X) = 1. Thus L is a p'-


group and there exists a unique X-invariant Sylow 2-group T of L. ThenM = NG(X) ~ N(T), so M ~ MT < G, a contradiction.

Let n: NG(L) --. N G(L )/CD(L) be the natural map, and define nf and 'G'f asin Theorem 1; that, n; is the set of complements N to D in G withAutN(L) = AutG(L), and 'G'; is the set of orbits of G on ,,;. Observe that byTheorem 2, the map

is a bijection between 'G'; and the set of Ln-classes (Yn)L" of complementsYn to Ln in NG(L)n such that Inn(Ln) ~ Auty,,(Ln). By definition nf £ Jl~,

and conversely a member X of n; is contained in nf if and only if X iscontained in no member of n!- Thus the conclusion of the first paragraph ofpart (C)(1) of Theorem 1 holds. We postpone the discussion of the secondparagraph until the next section.

The next lemma handles ~r(6.4) (1) If ME i then there exists FEc,?*(G,L) such that

M n Dr E,~ and M n D is the direct product of the M-conjugates ofMnD r ·

(2) The map M GH (M nDr)D r is a bijection of'G'i with

U 9 D /NG(D r »·rE.1'·(G,L)

(3) I~i I~ I3;' *(G, L)IIOut(L)I·

Proof If M E ni then 1 =1= M n D, so M n D projects nontrivially onsome member of .1. As M (\ D <J M and M is transitive on .1, M (\ Dprojects nontrivially on L. Then 1 =1= AutMI\D(L) <J AutG(L), so asL = F*(AutG(L», the projection of M nD on L is surjective. So 1.5 impliespart (1). Indeed 1.5, 1.3, and a Frattini argument yield part (2).

To prove (3) we may assume Y = 9 D /Nc (D r » is nonempty for someFE ,?*(G, L), and it remains to show leYI ~ IOut(L)I. But this follows from5.15, since by minimality of rc, NG(D r ) is primitive on F, so Nc(L) ismaximal in Nc(Dr ).

Finally, to complete the proof of Theorem 1, we deal with 'G' j.

(6.5) The map N C --. AutN(L)Inn(L) is a bijection of 'G't with the set ofconjugacy classes of maximal subgroups of AutG(L) not containing Inn(L).

Proof Let X = NG(L) and X* = X/CG(L) = AutG(L). By 6.2, NM(L) * ismaximal in X* if ME"'3' and M n D is the direct product of the Mconjugates of M n L, so the map

q>: N C H (NnX)*x'


takes qj'j into the set Y of conjugacy classes of maximal subgroups of X*which do not contain L *. If y*X* E Y then L n Y =1= I, X = LNxCL n Y),and Nx(L n Y) is maximal in X by 6.3. Let n be the set of subgroupsTIKe<1 WK , where WK E (L n y)G E K, and let WEn. By 1.3 and a Frattiniargument, G = DNG(W). A second application of 1.3 showsr = {En: z n L I is invariant under V = CG(L) and Cn(L) is transitive onr, so choosing W n L = Y n L, we have V = Cn(L) N v(W). ThusNxCL n Y)* = Nx(W)*. Hence NdW)G E ~t is mapped onto Y*x' under'fJ, so 'fJ is a surjection. By 1.3 and 6.2, 'fJ is an injection.

Notice we have completed the proof of Theorem I, modulo the proof ofTheorem 4. Theorem 4 is established in the next section.


In this section we assume the hypothesis and notation of Theorem 4. Weinvestigate which members of n; are nt'. In general nt is properly containedin n; ; for example, the argument in 6.3 shows:

(7.1) If Ooc/GjD) =1= I then ~~ is empty.

Recall Ooo(H) is the largest normal solvable subgroup of a group H. IfG = A 5 wrQ S where S is a split extension of E 16 by A 5 and n is of order 16,then by 7.1, ~t is empty, while by Theorem 2 n; is nonempty. On the otherhand there are many examples of groups G with ~t nonempty, as the nextlemma shows.

(7.2) If no subgroup of GjD properly containing NG(L)jD has ahomomorphic image whose generalized Fitting subgroup is isomorphic to L,then ~t =~;.

Proof. Suppose TE n; and T <XE n. Then X = T(Xn D) withX n D =1= I, so X E nf, By Theorem 1 there is r E .9*(G, L) and A E J;,such that A=XnDr . Then U=Nr(L) < V=NAA) and Inn(L)~

Autu(L) = Autu(A) ~ Autv(A), so F*(VjCv(A» ~L, contrary to thehypothesis of this lemma.

Observe that if G ~ A n wr'l An + l' where n is of order n + I, then by 7.2,~ t is nonempty.

(7.3) Let L be normal in a group H, K = CH(L), r the set of all normalsubgroups J of H contained in K with HjJL ~ HjK, 1= nJer J, andH* = HjI. Let :4' be the set of conjugacy classes xH of complements X to Lin H with H = XK. Then


(1) K* is the direct product of components Kt, 1<i <n, isomorphicto L and normal in H*. Moreover F = IHi: 1<i <n I, where

Ht = TIui K{

(2) Let ,SB=U7=1f?(LKi).(H*). Then for XHEsl', (XnLK)*=

H;* X (X n LKi )* for some i, and the map X H~ (X n LK;)*LK is a bijectionof sl' with .'15'.

(3) c>t;= IXH:Hi=XnKl is of order 0 or ICOUt(L)(OutH(L))I, soIc'Xl' I = m ICout(Ll(OutH(L))1 where m is the number of i, 1<i <n, with0; (LKj).(H*) nonempty.

Proof Let Q s:; F be of maximal order k subject to X = K/I ll =Xl X ... X X k , with Xi ~ L, I Q= nJEQ J, and Q = IHi: 1<i <kl, whereiii = f1*; X j ' Let J E F. I <l X with K/1 ~ L, so either I = X or x/I~ K/Jand hence J = Hi E Q; we may assume the former. Let e= IJl U Q andKa = K/lo. Then Ka =Ja X (IQ)a with Ja ~ X and (IQ)a ~ K/1~ L. Butnow the maximality of k is violated.

So (l) holds. Let X H E s/. Then X n KL is a complement to L in KL, soX n K <l LX = H, and as H = XK, (X n KL)/ (X n K) is also a complementto K/(X n K) in KL/(X n K). Thus LK/(X n K) = L(X n K)/(X n K) XK/(X n K) ~ L X L, so that part (1) and 6.1 imply parts (2) and (3).

(7.4) Assume FE !?*(G,L) and ADr E f?D/NG(Dr)). Then

(I) DCG(A) is normal in N G(Dr) and is the kernel of the mapNG(Dr ) = DNG(A) ~ NG(A)/ND(A) ~ Out(A).

(2) NG(L) nDCG(A) = DCG(L).

(3) Either NG(L) DCG(A) = NG(Dr) or DCG(A) = DCG(L).

(4) DCM(A) <l NG(Dr ) ifG = MD.

Proof If G=MD then DCM(A) <lDNM(A)=DNM(Dr)=NG(Dr), so(4) holds. (1) is trivial once we observe AutND(A)(A) = Inn(A).

D<NG(L), so NG(L)nDCG(A)=D(NdL)nC(A)). As FE!?*(G,L),NG(L) <NdDr). As ADr is NG(Dr)-invariant, DCG(L) = D(N(A) n DC(L)).The projection map A ~ L is N G(A) n N G(L )-equivariant, so N G(L) n C(A) <CG(L), and as DC(L) induces inner automorphisms on L, N(A) n DC(L)induces inner automorphisms on A so N(A) n DC(L ) <N(L) n A CG(A) <N(L) n DCdA). Therefore (2) holds.

As FE L'?*(G, L), NG(L) IS maximal in NG(Dr ). Thus asDCG(A) <lNG(Dr ) by (1), either NG(Dr)=NG(L)DCG(A) or DCG(A)<N G(L), and since it remains only to prove (3), we may assume the latter.Then DCG(A) = DCG(L) by (2), so (3) holds.

For FE !?*(G, L) let n!,r consist of those N E n! with N n Dr E ~. By6.4, nt is partitioned by (n!'r: FE ,'?*(G,L)).


THEOREM 7.5. Let M E n~ and set E = DCM(L). Let r E 5'*(G, L) withn{r nonempty. Then M is contained in a member of"t,r if and only if either

(a) E and DCG(L) are normal in NdDr), or

(b) the section NG(L)/E has a normal complement in NdDr)'

The proof of Theorem 7.5 involves a series of lemmas.

(7.6) If M is contained in a member of -4.1' then (a) or (b) holds.

Proof. Assume N E nt.r with M ~ N, and set A = N n Dr' ThenNM(L) ~ N G(A) so that the projection A -> L is N M(L )-equivariant. HenceCM(L) = CM(A) n N G(L) and therefore

(7.6.1) DCM(A)nNdL) = DCM(L)=E.

Let x E M n DCdA). As M E n~, Inn(L) ~ AutM(L) and then from theN M(L )-equivariance also Inn(A) ~ AutN ,\1(L)(A ). So there exists y E N..,(L)withy-1xE CM(A). Thus MnDCG(A)~NM(L)CM(A),do

(7.6.2) DCG(A) = D(Mn DCG(A)) ~ NM(L) DCM(A).

By 7.5.3 either DCG(A) = DCG(L) or NdL) DCdA) = NdDr)' In thefirst case DCG(L) <J N G(Dr ) by 7.4.1, and also DC,~t<A) ~ DCdL) <; N dL),so by 7.6.1, DCM(A) = DCM(L). So by 7.4.4, DCM(L) <J NG(Dr ). Thus (a)holds.

So assume the second case holds. Then by 7.4.4 and 7.6.2,DCM(A)<JND(Dr)=NG(L)DCM(A), so by 7.6.1 DCM(A) is a normalcomplement to N G(L )/E in N G(D r ). Thus (b) holds, completing the proof ofthe lemma.

Because of 7.6 we may assume during the remainder of the proof ofTheorem 7.5 that (a) or (b) holds; to begin assume (b) holds. Let K be thenormal complement guaranteed by (b). Then NdDr) = KNG(L) andK n N G(L) = DCM(L), so the map

(7.7) N G(Dr ) -> KNdL)/K -> OutG(L) -> Out(L)

extends the conjugation map NdL ) -> OutG(L) -> Out(L). Therefore byTheorem 5 there exists AD E QlD (NdDr)) for which the map in 7.7 is

r

Hence by 7.4.1 the kernel of this map is DCdA), while it is evident from 7.7that K is contained in the kernel. Thus K <; DCdA), so as D ~ K we have

(7.8) K = DCK(A).


Dr = ACD/L), so AD =ADr=AcDr(L). Then as NG(Dr ) acts on AD, by aFrattini argument NM(L) CDr(L) = UCDr(L) for some subgroup U of NG(A).As CDr(L) n N(A) = 1, U is a complement to CD(L) in NM(L) CD(L).

Let V = UCK(A). Then V <N(A). CM(L) CD(L) = C(L) nNM(L) CD(L) =C(L) n UCD(L) = Cu(L) CD(L), so DCu(L) = DCM(L ) = E. As K is acomplement to NG(L)/E, NK(L) = E. Therefore:

(7.9) NK(L) = DCu(L).

NdL) = DNM(L) and NM(L) CD(L) = UCD(L), so:

(7.10) NdL) = UD.

Therefore NK(L) = K n UD = CK(A)D n UD by 7.8. Then by 7.9:

(7.11) DCu(L)=UDnCK(A)D.

U normalizes A and L and hence commutes with the projection A -+ L, soCu(L)=Cu(A). By 7.9, Cu(L)<K, so:

(7.12) Cu(L) = Un CK(A).

From 7.11 and 7.12 we obtain:

(7.13) (Un CK(A))D == UD n CK(A)D.

Let a:V-- VD/D be the natural map. If uEU and kECx(A) withukEker(a) then ua=(ka)-IEUanCK(A)a==(UnCK(A))a by 7.13. Soas Un D = 1, u E CK(A). Thus uk E CK(A) so we have shown:

(7.14) ker(a)< CK(A).

Therefore V n D = CK(A ) n D == CD(A) = CD (Dr)' Also by hypothesisNG(Dr ) = KNG(L), while KNG(L)=KUD by 7.10, so by 7.8, NG(Dr )=DUCK(A) = DV. Hence:

(7.15) V/CD(Dr ) is a complement to D/CD(Dr ) in NG(Dr)/CD(Dr ).

As a consequence of 7.10 and 7.15 we have:

(7.16) UC(Dr ) = Nv(L).

From 7.16, Nv(L)CD(L)=UCD(L), while UCD(L)=NM(L)CD(L) byconstruction. Thus N v(L) CD(L) = NM(L) CD(L), so by 7.15 and Theorem 2,NM(Dr ) CD(Dr ) is conjugate to V in NG(Dr ). Therefore NM(Dr ) acts onsome member of AD, and hence we may assume without loss of generalitythat NM(Dr ) <NG(A). Hence M acts on B = (AM) and BE @D' ThusBE @(G) and M <NG(B) E nt.r'


This completes the analysis of case (b), so during the rest of the proof ofTheorem 7.5 we may assume (a) holds. Let K=CG(L), X=NM(L)CD(L),and Y=Knx. As ME n;, X is a complement to L in NG(L) withNG(L) =XK. By hypothesis E <J NG(Drr~ NdL), so Y = E nK <J NG(L).We now apply 7.3 with N G(L) in the role of H, observing Y is in the set r of7.3 because AutG(L) ~ X/Y (as ME ?tD. It follows that:

(7.17) Y<JNG(L), LK/Y=(LY/Y)X(K/y) with the factors NG(L)isomorphic, and (X n LK)/Y is a NM(L )-invariant full diagonal subgroup.

Notice Y= CD(L) CM(L), LY=E, and LK=DCG(L). By hypothesis LKand E are normal in NG(Dr ). Let Z = NG(Dr ) n C(LK/E). ThenNdL) ~ NG(Z) and as NG(Dr ) is primitive on r, either Z ~ NG(L) orNdDr)=ZNG(L). L~LK/E=F*(NG(L)/E), so ZnNG(L)=E, andhence in the latter case Z is a normal complement to N G(L)/E in N G(Dr ).But then we are back in case (a), so we may assume Z ~ N G(L). Hence:

(7.18) E = NG(Dr)n C(LK/E).

By 7.17, L is NG(L)-isomorphic to LK/E, so the map

(7.19) NdDr ) ...... OutN(Dr)(LK/E) ...... Out(L)

when restricted to N G(L) is equal to the conjugation map N G(L) ...... Out(L).Therefore by Theorem 5 there is a class AD E @D (NG(D r )) such that the

rkernel of the map in 7.19 is the same as the kernel of the map NG(Dr ) ......OutN(A)(A) ...... Out(L). By 7.18 the kernel of the map in 7.19 is E, while by7.4.1 the kernel of the second map is DCG(A).

Thus we have shown:

(7.20) E = DCG(A).

As in case (a), there is a complement U to CD(L) in X which acts on A.Let Uo=UnLK. Then:

(7.21) XnLK= UoCD(L).

We conclude from 7.17 and 7.21 that:

(7.22) UoY/Y is a NM(L)-invariant full diagonal subgroup of LK/Y=LY/YX K/Y.

LK = A (X n LK) by 7.22, since LY = A Y. Thus

NLK(A) = (A(XnLK)) nN(A)

= A (N(A) n X n LK)

=A(N(A)n UoCD(L)) by 7.21

=AUoCD(Dr )·


(7.23) NLK(A)=AV.

CG(A) ~ E ~ NdL) by 7.20, so the projection A --+ L commutes withCdA) and thus:

(7.24) CG(A) ~ CdL) = K.

Next NLK(A) induces inner automorphisms on A, so NLK(A)~ACG(A),

and the opposite inclusion follows from 7.24, so:

(7.25) NLK(A) =A X CG(A).

Let a: ACdA)--+A be the projection map. For vE V, v=a(v)k for somek E K by 7.24, so as LY = AY we conclude:

(7.26) Ya(v) is the projection of Yv on LY/Y for each v E V.

Let us examine the map in 7.19 more closely. Given an isomorphisma: T --+ S let a*: Out(T) --+ Out(S) be the induced isomorphism defined bya *(;iT 0 qJ) = lis 0 a 0 qJ 0 a -I, for qJ E Aut(T) and liT: Aut(T) --+ Out(T) andlis: Aut(S) --+ Out(S) the natural maps. We chose A so that the diagram

OutN(Al(A)

/ i ~II

NdA ) I Out(L)10'

~I

~II

OutN(Al(LK/E)

commutes, where n: A --+ L is the projection and (using 7.21) a: LK/E --+ L isthe composition of the inverse of the projection V/ Y--+ K/ Y ~ LK/E with theprojection V/Y--+LY/Y~L. Let O=a 0 n- I

; then we can fill in 0* in thediagram and still keep it commutative. Observe that by 7.26 and thedefinition of ():

(7.27) O(Ev) = a(v)for each v E V.

Let Y E NdA). Then y acts on ACdA) = AV by 7.23 and 7.25. Thus forv E V, vY = y(v)P(v) for some y(v) E A and P(v) E V. Notice (EvY = EP(v).Moreover from the commutativity of the diagram, there exists x E A suchthat ()( (Ev Y X

) = ()(Ev y x for each v E V. As x E A ~ E, (Ev yx =(EP(v)y=EP(v), so by 7.27, a(p(v»=a(vyx=a(vYX

) for all vE V.

481/92/1-6


Next vYx = (y(v) P(vW = y(vY [x, P(V)-IJ P(v) with o(v) =

y(vY [X,P(V)-l] EA. So a(fJ(v» = a(vYX) =a(o(v)p(v» = o(v) a(fJ(v», and

hence o(v) = 1. So vyx = P(v) E V, and therefore:

(7.28) NG(A) ~ NG(V)A.

Let W=NG(A)nN(V). By 7.28, NG(A) = WA. As LY=AY, 7.22implies NA(V) = 1, so W is a complement to A in NG(A). Thus as NG(Dr ) =NG(A) Dr> we conclude:

(7.29 W/CD(Dr ) is a complement to D/CD(Dr ) in NG(Dr)/CD(Dr ).

Recall that M acts on A and observe that U acts on Uo, and hence also onV. So U ~ W. Thus CD(L) Nw(L) = CD(L)U = X = CD(L) NM(L), soapplying Theorem 2 to NG(Dr)/CD(Dr ) and using 7.29, we conclude thatN M(Dr ) is conjugate in N G(D r) to a subgroup of W. Hence without lossNM(Dr ) ~ W. Then M normalizes the direct product T of the M-conjugatesof A, so M ~ N G(T) E ,ni.r, completing the proof of Theorem 7.5.

Weare now in a position to prove Theorem 4, and hence complete theproof of Theorem 1. Let ME ,n;. As we observed during the proof of 7.2,ME,n{' if and only if M is contained in no member of nj'. Hence as ni ispartitioned by (,ni,r:rE,'?*(G,L», ME~,t if and only if for eachrE'?*(G,L), M is contained in no member of ~'i.r' Therefore Theorem 7.5implies part (1) of Theorem 4.

Let H = NG(L) and n: H ---. H/CD(L) the natural map. By Theorem 2,p: M ---. N M(L)n is a bijection of ,n; with the set ,(/' of complements Yn to Lnin Hn with Inn(L) ~ Auty(L), and f1 induces a bijection f1 *: 'it?; ---. 13 of thecorresponding orbits of G and H. By part (1) of Theorem 4, p(,nn consistsof those Yn E Y with LCy(L) E g. Adopt the notation of 7.3 with (Hn, Ln)in the role of (H, L) in that lemma. Further choose notation so that .~ isnonempty if and only if 1 ~ i ~ m. Observe that {LH j : 1 ~ i ~ m} is the setof normal subgroups of H satisfying conditions (i) and (ii) of the definitionof g and LCy(L)=LHi for each (Yn)HE.W;. (Hi is defined in 7.3.)Therefore 7.3.3 implies part (2) of Theorem 4.

This completes the proof of Theorem 4.We close this section with another parameterization of 'it? 7 and 'it?;.

Namely, in the notation of Section 2, we prove:

(7.30) Let.o1' be the set of classes N L of complements N to L inNG(L)/CD(L) with Inn(L) ~ AutN(L), let 17: 'if?' ---. JlI' be the map supplied byTheorem 2, let 11: JlI' ---.%omtG(L)(DCdL)/D, L) be the map supplied by 2.2,and let a = 11 0 f1 be the composition of 17 and p. Then

(1) a defines a bijection 'it?; ~%omNG(L)(DCG(L)/D, L).


(2) a ('It'n consists of those orbits qJL such that the inverse image inDCdL) ofker(qJ) is in g.

Proof By Theorem 2 and 2.3, f.J and YI are bijections, so a is a bijection.By Theorem 4, M G E 'It" is in 'It' * if and only if E = DCM(L) E g. ButEID = ker(YlN) where N = NM(L)Cn(L)/Cn(L), and YIN is defined in 2.2 andN = f.J(M), so (2) holds.

APPENDIX: CORRECTIONS TO REF. [4]

In an appendix to [4] two theorems on maximal subgroups were statedand proved in outline. The following corrections should be made in thestatements of these results: In the first theorem "Let H" at the beginningshould read "Let H =1= 1," and the expression "prime cardinality" in part (a)should be "cardinality =1= 1." In part (e) of the second theorem "p" should be"k"; also "subgroup" in the first line should be "proper subgroup."Corrected statements of these theorems are given below.

THEOREM. Let H =1= 1 be a subgroup of a finite direct productG = TIiEl Gi of isomorphic nonabelian simple groups. Then the transitivepermutation representation of G on GIH extends to a primitive permutationrepresentation of some group in which G is the socle if and only if either

(a) there is a partition c9' of I into subsets of equal cardinality =1= 1with H the direct product TIsE9'Ll s of full diagonal subgroups LIs of thesubproducts TIi<S Gi, or

(b) the subgroup H is a direct product TIi <;l Hi where Hi is asubgroup of Gi which is an intersection Gi n iii for some maximal subgroupiii of a group Gj with Gi <;; Gj <;; Aut Gj • Also, for each pair i, j of indicesthere must be an isomorphism of Gi with Gj carrying Hi to H j •

THEOREM. Let M =1= cw" be a proper subgroup of the symmetric group S n'

Then some conjugates ofM is contained in one of the subgroups listed below.Here 1 < m < nand p is prime.

(a) SmXSk,m+k=n (intransitive),

(b) S mwr S k' mk = n (imprimitive),

(c) Sm wr Sk' mk= n, m ~ 5 (product action),

(d) V.GL(V), pk = n = IVI; Va vector space over GF(p),

(e) (G wr S k)Out G, IG Ik -1 = n, G a nonabelian simple group,

(f) an automorphism group of a nonabelian simple group G < c#n'containing G and acting primitively (the full normalizer of G in S n)'


The group in (e) is the extension of G wr S k by the outer automorphismgroup Out(G) obtained from the natural extension Aut(G) of a diagonalcopy of G; this isotropy group is Aut(G) X S k' As mentioned in [4] thistheorem was obtained independently by Mike O'Nan.

The assumption H -=1= I in the first theorem is necessary, as the examples atthe end of the section Main Results show. A similar correction should bemade in Cameron's statement of this result. Theorem 4.1 in [I]. (Thepossibility GanN = I should be allowed.) However, this probably does notaffect the applications he discusses. It has no effect on the list above ofpossible maximal subgroups of the symmetric and alternating groups, sincethe groups arising this way are contained in subgroups of type (c).

REFERENCES

1. P. CAMERON, Finite simple groups and finite permutation groups, J. London Math. Soc. 13(1981), 1-22.

2. D. CURTIS AND I. REINER, "Representation Theory of Finite Groups and AssociativeAlgebras," Wiley, New York, 1962.

3. S. MACLANE, "Homology," 3rd corrected ed., Springer-Verlag, New York, 1975.4. L. SCOTT, Representations in characteristic p, Proc. Sympos. Pure Math. 37 (1980),

319-332.5. J. SERRE, "Corps locaux," Hermann, Paris, 1962.6. A. BOREL AND J. SERRE, Theoremes des definitude en cohomologie galoisienne, Comm.

Math. Helv. 39 (1964), 111-163.7. F. GROSS AND L. KOVACS, On normal subgroups which are direct products, preprint.8. F. GROSS AND L. KOVACS, Maximal subgroups in composite finite groups, preprint.

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Maximal Subgroups of Finite Groups - University of Virginiapi.math.virginia.edu/~lls2l/maximal-subgroups-of-finite-groups.pdf · MAXIMAL SUBGROUPS OF FINITE GROUPS (I) Determine ~t.G~HI(G,V)