The Abstract Groups Rm = Sm = (Ri Si)Pi = 1, Sm = T2 = (Si T)2Pi = 1, AND Sm = T2 = (S-i T Si T)Pi = 1

278 H. S. M. COXETER [Feb. 20,

THE ABSTRACT GROUPS

= Sm = (R* S'VJ = 1 , Sm=T2 = {8* Tf»i = 1,

AND Sm=T2= (<S-' T8* T)*J = 1.

By H. S. M. COXETER.

[Received 8 February, 1936.—Read 20 February, 1936.]

Page1. Preliminary remarks 2792. The notion of augmenting a group by adjoining the cyclic permutation of its

generators 2813. Application to groups generated by reflections 2814. Special cases 2835. A generalization of the symmetric group 2856. A generalization of the alternating group of odd degree ... ... ... ... 2887. A generalization of the alternating group of even degree 2918. Symmetry groups of the Cartesian frame 2939. Groups involving the fc-th power of the symmetric group ... ... ... 297

In this paper we investigate the groups

j=h 2,

= T2 = {S-* TS* T)pJ = 1

We prove that they are finite if tpi = 2 for every j < \m, and infiniteotherwise. For this purpose, we show how these groups can be derivedfrom groups generated by reflections. In an earlier paper we developeda geometrical method which enabled us to enumerate all finite groupsgenerated by reflections. The above theorem then follows by comparisonwith those results.

* Groups of the form Sm = T2 = (SJT)V = 1,

where the g's may be odd, present a far more difficult problem.

1936.] SOME ABSTRACT GROUPS. 279

The geometrical interpretation is especially applied to the case whenpx = 3 while every otherpt = 2. In this case the reflections can be regardedas transpositions of m symbols, and from this it easily follows that thegroup

=:l (j = 2, 3, ..., [\m]\ m>2)

involves the symmetric group of degree m as a factor group, namely as thequotient group of an infinite Abelian subgroup. This subgroup appearsin the geometrical representation as a group of translations. By mag-nifying these translations, we obtain a larger quotient group, which in onecase is related to the symmetry of the Cartesian frame (e.g. when m = 3,to the cubic system of crystallography). A more general application ofthe same principle is developed in § 9.

I should like to take this opportunity to express my gratitude to Dr.O. Taussky for her helpful and constructive criticism.

1. Preliminary remarks *.

We interpret the abstract definitions

(1.1) Rm = Sm = {Rj S>)VJ = 1

(1.2) 8m = T2 = (£> Tp> = 1 • (j = 1, 2, ..., [£m]; pi > 1)

(1.3) Sm=T2= {8-i TS* T)»J = 1.

as implying that the period of Rj Sj or {S> Tf or 8~} T8j T (respectively)is precisely pi (and not merely a factor of pt). The condition pf> 1 isinserted because, as we shall see in §3, them's can all be arbitrarily assignedso long as they are all greater than 1. To see what happens in the contrarycase, we may use the following argument.

We first extend the definition of Pj (as a period), beyond the given rangeof j . The new values are determined by the formulae

Now,3^= 1 implies that^,=pAj±iforalliandA, and pi=pk= limpliesthatPKJ±IIJI= 1 for all A and /x. Let I denote the first (positive) value of̂ ' forwhich Pj= 1. Then Vis a factor of m (since pm = 1 ) , and all other suchvalues of j are just the multiples of I. (As an extreme case, we may haveI = 1, and all pt = 1.)

* The reader who is in a hurry may turn at once to. § 2. .


Thus the groups (1.1), (1.2), (1.3), without the restriction ipi > 1, are

8m=T2=(SlT)2=(8iT)2v<=l\ (i=l, 2, ..., [#]; l\m\ pt>l),

j=l, 2, ...,

cr, with a change of notation,

(1.4) Rnm = 8nm =Rm8m=(Ri 8')** = 1

(1.5) snm=T2 = (8m T)2 = (8* T)2PJ = 1

(1.6) Snm =T2 = 8~m TSm T — (S~j TS> T)pJ = 1

In each case, 8m generates a cyclic self-conjugate subgroup of order n.The quotient groups are (1.1), (1.2), (1.3), respectively. In (1.4) and(1.6), and in (1. 5) when n = 2, this cyclic subgroup is central.

When m and n are co-prime, we can exhibit (1.6) as the direct productof (1.3) and the cyclic group of order n. To do this, we select integersfx., v, such that

fin—vm = 1,

and let Sf = S*11 = Svm+1.

Since S 1 belongs to the central, the operators 8' and T of (1.6) generate(1.3) (with 8' in place of 8). Every operator of this subgroup of (1.6),of index n, is permutable with every operator of the subgroup of order ngenerated by 8m. Hence (1. 6) is their direct product.

Similarly, writing R' = i?Mn = R"m+1, we see that, when m and n areco-prime, (1.4) is the direct product of (1.1) and the cyclic group of ordern. Also (1.5), with m odd and n = 2, is the direct product of (1.2) andthe group of order 2.

Since, in (1.4), (1.5), and (1. 6), respectively,

R'-18f = R~18.82™, S'-1 TS'T = S~2vm. S-1 T8T = S-1 T8T. 8-*™,

and S'T=8T.Sim,

all these statements continue to hold when we modify the groups byadjoining the relation

(i?-i/Sf)*"«=l to (1.1) and (1.4),

(S-1 TST)** =1 to (1.2) and (1.5),

(ST)^=l to (1.3) and (1.6),

provided that the value of q is such as to render these relations consistent


with the original ones. (E.g., when px = 3 while every other p} = 2, wewrite # = #1—1.)

2. The notion of augmenting a group by adjoining the cyclic permutation ofits generators.

If an abstract group G is formally symmetrical in its I generators, wecan derive a larger group F, in which G is a self-conjugate subgroup ofindex I, by adjoining the cyclic permutation of the generators; i.e., weintroduce a new abstract operator T, such that

Ri=T-iRlTi, Tl=l,

and so obtain a group generated by Rt and T. The operator T may trans-form G according to either an inner or an outer automorphism. In theformer case, V is the direct product of G and the cyclic group of order I.(For, if t is the operator of G that has the same effect as T, then T~xt ispermutable with every operator of G.) In the latter case, if Tk (but nolower power of T) transforms G according to an inner automorphism,F is the direct product of a subgroup of the holomorph of G and the cyclicgroup of order l/k. Of course it may happen that k = I.

For instance, let G be the group (1.1), which is formally symmetricalbetween E and 8. By writing

R=TST, T2=l,

we derive the group (1.2), whose order is therefore twice as great. Now,the symmetry between R and 8 is preserved if we replace R by R*1. Hencethe group (1.3) has the same order as (1.2). In fact, (1.2) and (1.3)have a common self-conjugate subgroup of index 2, namely (1.1).

Similarly, (1.5) and (1.6) have the same order, each containing (1.4)as a self-conjugate subgroup of index 2.

These statements continue to hold when we adjoin extra relations asat the end of §1.

3. Application to groups generated by reflections.

In an earlier paper* it was seen that the abstract relations

r A2=i (»=1,2, ...,m),(3.1) \

{(RiRi)k<j=l (».= 1, ..., TO-1; j = i+l,...,m; ka>l)

* Coxeter, 3. (Dr. Du Val has pointed out that Lemmas 1 and 2 are unnecessary. Inthe proof of Lemma 3, the invariant positive definite form can immediately be used toestablish a Euclidean metric.)


are consistent, and that the group defined by them is finite for certainspecified values of the integers ku. We shall call it a g.g.r. (" group gen-erated by reflections "). At present we are concerned with the specialcase when

hi=Pi-v Pi=Pm-i>

so that the group has the cyclic permutation of the R's as an automorphism.The representative graph generally consists of an m-gon with its sidesmarked pv its first diagonals marked p2, and so on. But in all the inter-esting cases this will be simplified by most of the p's being equal to 2.

The above mentioned results show that the group

B?=l (»=1 , 2, ...,ro),(3.2)

{R.RM)PJ= 1 (i= 1, ..., m—1; j= 1, ..., m—t; pf=Pm-,>l)

is finite only when pi = 2 for all j < \m. When every pi = 2, it is []m,the Abelian group of order 2m and type (1, 1, . . . ) ; the graph consistsmerely of m separate dots. When m is even and every Pj = 2 exceptPim = P^ *n e g r o uP is [p]*m, of order (2p)im, the direct product of \m dihedralgroups; the graph consists of \m pairs of dots, each pair being linked,and marked with the number p.

Let S denote the cyclic permutation of the R's, so that

Ri=8-iEm8it Sm=l.

The relations (3.2) become

*>j=l (j=l, 2, . . . ,

Hence, if we write T in place of Rm, the group derived from (3.2) byadjoining 8 is

(1.3) Sm = T2 = (8-* TS* T)PJ = 1 (j = 1, 2, ..., [|m]; p, > 1).

This group, and therefore also (1.2), is

(i) of order m. 2m when every pi = 2,

(ii) of order m(2p)^m when m is even and every pi = 2, except p±m = p,

(iii) infinite in all other cases*.

The subgroup (1.1) has half this order, in each case; (1.5) and (1.6)have n times this order; and (1.4) has \n times this order.

* Although, (i) and (ii) are trivial, I believe that it would be really difficult.to prove(iii) without the earlier paper's appeal to geometry.


4. Special cases.

I t is perhaps worth while to write down explicitly the finite groups ofthe form (1.1), (1.2), (1.3) :

of order m. 2m-1;

(A O\ ,Qm 7*2 _

of order m.2m;

(4.4) Rm=:Sm= (Ri Si)2 = (R*m S*m)*> = 1

(j = 1, 2, ..., *w*— 1; m even),of order

(4.5) | ( j , , . . . , * ;

(4.6) 8m = T2 = (£-> T£>" T)2 = (/S*m T)2P = 1J m even)»

of order m(2p)*m.The group (4.3) has an obvious representation as a permutation

group of degree 2m:

8={alat...am)(b1bt...bm), T= (o^6m).

In fact, 8~* TS> T = (a3-6,)(am6m).

The 2m symbols a,j} bt can be regarded as the vertices of the cross-polytopejSm, i.e., points at a fixed distance from the origin in both directions alongall the axes of an m-dimensional Cartesian frame. 8 is just the cyclicpermutation of the axes.

The operator

(8Tr=(alb1)(a2b2)...(ambm)

generates the central (of order 2). Thus the central quotient group*

(4.7) Sm=T2= (ST)m={S-iTSjT)*=l {j=l, 2, ..., [$(m— 1)]}

is of order m. 2m~1. When m is odd, the relations (4.7) are satisfied bythe operators S, T{ST)m of (4.3); hence the group (4.3) is then thedirect product of (4. 7) and the group of order 2.

* It is easily seen that, when m is even, the relation

(S-i-'-TS^T)* = i

is a consequence of the rest. Cf. Coxeter, 4.

284 H. S. M. COXETEB [Feb. 20,

When m = 3, (4.1) and (4.7) reduce to the tetrahedral group, while(4.2) is octahedral, and (4.3) pyritohedral.

When m = 4, (4.4), (4.5), and (4. 6) become

of order 16p2.

B* = S4 = (RS)2 = (R2 8*Y> = 1 * , of order 8p2,

S*=T2 = {ST)* = {S2 T)2* = 1 f, ]

S*=T2 = (S-1 TST)2 = {S2 Tfv = 1J, J

When m = 5, (4.1) and (4.2) become

R* = S5= (RS)2 = (R2 S2)2 = 1, of order 80,

S*=T*={ST)*={S*T)*=1, of order 160.

The conclusion of § 3 enables us to assert that the groups

R* = S* = {B8y* = (JB2 S2)v> = 1,

S*=T2= (8T)2*» =(S2 T)2»* = 1,

S*=T2= (S-1 TST)?} = (S* T)2** = 1,

are infinite if pr > 2 and jp2 > 1, and that the last two of these are stillinfinite if px — 2 and p2 > 2.

When m > 5, (1.1), and (1.2) involve the relations (R383Y>» = 1 and(<S3T)2*S= 1 (respectively), and by omitting them we tacitly make p3

infinite. Hence the groups§

(4.8) . Rm=8m

(4.9) Sm = T2 = (/ST)2^' = (82 T)*P* = 1

are infinite if TO > 5 (and plt p2 > 1).

* Burnaide, 1; III, a = 6 = p.t Ibid., a = 2p, 6 = 0.X Since

{FT)* = S-tT.TS-iTS.ST = S^T.S^TST.ST = {S~l T)* (ST)*,this is a special case of the group

S* = T* = (S-* TST)* = (S-1 T)tb (ST)U = 1,

of order 8(6*+c*). ^ee Coxeter, 6, 65; Sinkov, 1, 81.§ When pl =2, the expression R*S* can be replaced by a commutator, since

i?1 S* = R8-1 R-1 S.


5. A generalization of the symmetric group.

The most interesting infinite group of the form (1 .3) is

(5 .1) Sm==T2= (S-l TST)3 = (£-; TS> T)* = l

(j=2, ..., [£m]; m > 2)*r

which is derived from the g.g.r. whose graph is an m-gon. This g.g.r.fcan be generated by reflections in the primes

Xl = X2> X2 = XS' •••> xm-l ==xm> Xm == xl~^~ *

of Euclidean m-space. But the group operates essentially i n m - 1 dimen-sions, since all the primes are perpendicular to

These generators are transpositions of the coordinates, combined! in onecase with a translation. Hence all the operators of the g.g.r. are operatorsof the symmetric group combined with translations, and S is thecyclic permutation (xx x2... xm) combined with a translation. TakingT to be the transposition {xxx2), i.e. the reflection in ^ = ,r2. ST is(x2...xm) combined with a translation, and consequently (ST)"1'1 isa pure translation^ The operator (ST)™'1 and its conjugates generatean infinite Abelian self-con jugate subgroup of (5.1), whose quotient group

(5.2) Sm=T2= (ST)™-1 = (S-1 TST)3 = (S~j TSj Tf = 1

(j=2, ..., [hn]; m>2)

is the symmetric group (of degree ra)||. Since the number of dimensionsis m—1, the longer translation (ST)7*™-1) and its conjugates generate asubgroup of index nm~x in the above Abelian group, and we have the more

* De S^guier, 1, 261 (4). When m = 2, (5.1) reduces to the dihedral group [6], oforder 12.

f Coxeter, 1, 162 (§17.9).I By " combined with " I first mean " transformed by." But transformation by one

translation is equivalent to multiplication by another.§ Cf. Coxeter, 6, 72 (Lemma 6.2). {ST)m~x cannot be the identity, since then the

groups (5.1) and (5.2) would be the same, whereas we know that they are infinite andfinite, respectively.

|| Moore, 1, 358. As a g.g.r., this is [3'"-»].

-SO H. S. M. COXETER [Feb. 20r

general quotient group*

(5.3) Sm=T2= {8T)n<m-v = (S-1 TST)3 = (£-> T8* Tf = 1

O'=2, ..., flm]; m>2),Oof order w7"-1™!.

This is derivable from the larger group

(5.4) Snm = T3 = (ST)71^-1* = S~m TSm T = {S-1 TST)*

as the quotient group of the cyclic subgroup (central) generated by Sm.The group (5.4) is thus of order nmm\. We proceed to prove that it isrepresentable as a permutation group of degree nm in the form

(5.5)

These permutations satisfy all the relations, since

S-' TS* = ( l m li+a)(2i+r2J+1)..

and Sm = s1s2...8m,

where s( = (1,2,-... nt).

We have also {ST)m~x = s2... sm,

whence

(5.6) S

and S1~m(ST)m-1 = Ss-1

= t, say.

The permutations i and T generate the symmetric group on the m suffixes.Since

8 = tav

the group generated by the permutations 8 and T coincides with thatgenerated by sx, t and T; it is therefore of order nmm\, its general operatorbeing expressible in the form

(5.7) ( )«}«&...«!; ( 0 < ^ < n - l ) ,

* In every numbered abstract definition, until the end of § 6, we shall forgo repeatingthe statement of the range of j and the restriction on m.


where () stands for any permutation of the suffixes, i.e. any operation ofthe symmetric group generated by t and T. Hence the given permuta-tions generate the whole group (5.4) (and not merely a factor groupof it)*.

Whenw = 2, sothat >S = ( l 1 l 2 . . . l m 2 1 2 2 . . .2 m )and T = (1112)(2122),the symbols 13- and 2̂ can be regarded as opposite vertices of the polytope/3m. The operators t and T permute the Cartesian axes, while s5 reversesthe sign of the coordinate xf. Thus

(5.8) S2m = T2 = (ST)*m-v = S~m TSm T = (8-1 TST)*

is the hyper-octahedral group f [3m~2, 4]; and

(5.9) Sm = T2 = {ST)*m-» = (S-1 TST)S = (S-f TS* T)2 = 1

is the central quotient group £[3m~2,4], which can be regarded as a g.g.r.in elliptic (m— l)-spacej.

When n > 2, the symbols

li> 2ls ..., n-^'y 12, 22, ..., n2; ...; lm, 2m, ..., nm

can be regarded as the vertices of m equal, concentric ?i-gons, lying inabsolutely perpendicular planes in Euclidean 2m-space. The operatorSf is then a rotation through 2TT/7I in the ̂ '-th plane, while t and T generatethe symmetric group on the m planes (preserving the sequence of thevertices of the polygons).

For the sake of closer analogy with the case when n = 2, we may workin complex m-dimensional space, letting hj (h ^n, j ^ m) denote thepoint whose j-th coordinate is e2nih/n, while the remaining coordinatesvanish. (The transformations are now unitary instead of orthogonal.)

* The group consists, in fact, of all permutations (of the nm symbols) comxautnt ivs with

(I121. . .n1)(l222. . .n2) . . . ( l , , ,2m. . .n, , ( ) .

Cf. Miller, Blichfeldt, and Dickson, 1, 20 (§ 8).| Young, 1; Coxeter, 2, 607. The symbol [31""2, 4] means (3.1) with

&i2 = &23 = ••• == im-Jipi-i = 3, &»,-im = 4, and every other k{j = 2 .

The isomorphism with (5.8) may be verified in a direct manner by writing

S = Rmi?,,,-! ... i?i, T = JBj,

whence Ri+i = S-* TSl (i < m— 1)

and Rm = SRXi?2 ... i2,,,-i

For the actual work involved, see Coxeter and Todd, 1, (7).j Coxeter, 2, 616.


When m and n are co-prime, the argument used in § I shows that (5.4)is the direct product of (5.3) and the cyclic group of order n. Regardedas a subgroup of (5.4), (5.3) is generated by S"m+1 and T. These can bewritten as permutations.

In particular, when n = 2 and m is odd, we have

or, by a change of notation,

S'=(a1a2...am)(b1b2...bm), T= ( a A ) ^ ) .

These permutations, when regarded as operating on the vertices of /?,„,belong to the g.g.r.* f̂ "1"3'1*1] (the complete symmetry group of the half-measure-polytope hym). This geometrical group, being of order 2m~17n!,can thus be identified with (5.3) (m odd, n = 2). On comparing thiswith our previous representation for (5. 3), we see that

(•3m-3,1.1] ̂ I [3m-2j 4] (m ^ d ) .

In § 8 we shall give a more direct proof of this result.Putting m = 3 or 4 in (5. 3), we obtain

£3 = T2 = (ST)2n = {S-1 TST)3 = 1 f, of order 6?i2;

S*= T*= (ST)*n = {S-1 TST)* = (S2 T)* = 1, of order 24w>.

6. A generalization of tlie alternating group of odd degree.

When n(m— 1) is even, the operators TS'1 T and S of (5 . 3) generate Jthe group

(6.1) B™ = Sm = {R-1 S)M™-v = (RSf = (R* iSP)a = 1,

of order \nm~xm\. Introducing a new T so that

R=TST, T*=l,

we derive the group

(6.2) Sm = T2 = (S-1 TST)Wm-» = (ST)* = (8* Tf = 1,

* Called 3 in Coxeter, 1, 151; 2, 607.L3 _j

f Sinkov, 1, 82.I Cf. §2, and Coxeter, 4.


of order nmr-1m\. Similarly, (5.4) leads to

(6.3) Rnm = 8nm = (R-1 S)Mm-» =RmSm= (RS)* = {R* &)* = 1,

of order \nm m!; and

(6.4) 8™ =T2= {IS-1 TST)M»-» = (S™ Tf = (ST)* = {8> T)* = 1,

of order nmm\.When n = 1 (and m is odd), (6.1) is [S"*-2]', the alternating group of

degree m, generated by the permutations

R = (a1a2amam-1...a3), S = (a1a2a3...am).

The operator T of (6.2) interchanges R and 8, and therefore (when n = 1)transforms the alternating group according to the automorphism

(a3am)(a*am-l) ••• (°|(m+Dat<m+5))>

which is an outer or inner automorphism according as £(w-f 1) is odd oreven*. Thus (6.2) with n = 1 is the symmetric group [3m~2], or the directproduct! [3OT~2]' X [ ], according as m = 1 or 3 (mod 4). We have shownelsewhere % that the relation {S-1 T8T)Z = 1 may be omitted from(5.2) [though not from (5.1)]. It follows that, when n=l, the relation(jR f̂)3 = l may be omitted from (6.1), and (8T)«= 1 may be omittedfrom (6.2). We have thus an elegant new definition for the symmetricgroup of degree m = 1 (mod 4):

(6.5) 8m=Ti = (8-1T8T)Um-1>> = (SiT)*=l [j = 2, ..., | ( m - l ) ] .

The above connection between the groups (6.1), (6.2), and (5.3),when n = l , continues to hold when n = 2. In this case (6.1), (6.2),(6.3), (6.4) become

(6.6) R»>=8m= {R-1 S)™-1 = {RS)* = (R> &)* = 1,

( 6 . 7 ) 8m = T2 = (S-1 T8T)™-1 = (8T)« = {8* T)4 = 1,

(6.8) R2m = 8*™= {R-18)™-1 = R"8m= (RS)* = (R* S')* = 1,

(6.9) 82m =T2= {S-1 TST)™-1 = (Sm T)2 = (ST)« = {S> T)*=l.

When m = 4, (6.7) and (5.9) are obviously identical. We proceed toprove the following theorem.

* I am indebted to Mr. P. Hall for this remark.| The direct product of the alternating group and the group of order 2.t Coxeter and Todd, 1, (5).

8SB. 2. vol.. 41. NO. 2112. U

290 H. S. M. COXETEB [Feb. 20,

When m~Q or 1 (mod 4), the groups (6.7) and (6.9) are the same as(5.9) and (5.8) respectively. When m = 2 or 3 (mod 4), (6.7) is the directproduct of (6.6) and the group of order two, while (6.9) is similarly relatedto (6.8).

Writing ap &,• for lp 2,- in (5.5), we represent the generators of (5.8)and (6.8) in the form

8 = {a1a2az...amb1b2b3...bm),

R=TS-1T={a1a2bm...b3b1b2am...a3).

The T of (6.9) transforms (6.8) according to the automorphism

or T" = K bx) (a2 b2) (a3 am) (a4 am_x)... (b3 bj (64 b^

Actually, T" is more convenient than T' for our present purpose, since,in the notation of (5.7),

T"=()slS2,

where, as a permutation of suffixes,

( £ , \m+2) (m even).= (3 , m)(4, m-1) . . .

( i + i | + f ) (m odd).

This is an odd permutation if m = 0 or 1 (mod 4), and an even permutationif m = 2 or 3 (mod 4).

Now, the operators of (6.3) are just those operators of (5.4) whoseexpression in terms of S and T involves T an even number of times. Since(5.4) contains all permutations of the suffixes, and since every transposi-tion is conjugate to (1, 2) — T, (6.3) must contain all the even permuta-tions. In particular, (6.8) contains all the even permutations. It alsocontains sx T~1s1 T = s ^ ; hence it does or does not contain T" accordingas () is an even or odd permutation. In the former case, T" is an innerautomorphism of (6.8), and (6.9) is a direct product. In the latter, T"is an outer automorphism of (6.8), but still belongs to (5. 8).

Our theorem is now proved, so far as (6 . 9), (6. 8), and (5. 8) are con-cerned. Consequently, the central quotient groups (6.7), (6.6), and(5.9) are related in the same manner.


The argument used in § 1 shows that, when m is odd, (6.8) is the directproduct of (6.6) and the group of order two, while (6.9) is similarly relatedto (6.7). Hence, when m = 3 (mod 4), (6.7) and (6. 8) are the same group.

The following particular cases of (6.1) and (6.2) seem worthy ofexplicit mention. (We put m = 3,4,5, in turn, and write 2q for n when-ever n has to be even.)

R3 = S*= (RS)* = (i?-1 S)n = 1*, of order 3^2,

S* = T2 = (ST)6 = (S-1 TST)n = 1 * , of order 6n2,

Ri = S*= (RS)* = (R2 S2)2 = {R-1 S)3* = 1, of order 96g3,

S*=T2= (ST)6 = (S2 T)4 = {S-1 TST)*a = 1, of order 192?3,

R^ = S5= (RS)* = {R* £2)2 = (R-1 S)2» = 1 , of order 60n*,

= (£2 T^ = (£-i TSTfn = 1 , of order I20rc4.

7. A generalization of the alternating group of even degree.

When m is even, the following remark can be made about all the groups(1.2), (4.7), (5.3), (6.2): every generating relation involves the gen-erators an even number of times. It follows that the operators of eachgroup fall into two classes, according to the parity of the "degree" oftheir expressions as products of S's and T's. (E.g., ST is o£ degree 2,S~2T of degree — 1.) The class of operators of even degree constitutesa self-con jugate subgroup of index 2. Since ST. TS = S2, this sub-group is generated by

U=TS, V = ST.

(It is symmetrical between U and V, the automorphism which interchangesthem being effected by T.)

When m = 4 or 6, an abstract definition for the subgroup is immediatelyobtainable by observing that

S2=VU, (S2T)2=VU2V, S*T=VUV, S~1TST=U-1Vi

S-2TS2T=U-1V-1UV.

Taking the four groups in the above order, we have, when m = 4 :

* Burnside, 1, II, IV; o = n, b = 0.t This is merely a special case of (4.8).

U2


of order 8p22 if pt = 2, infinite ifp1>2 (and p2 > 1);

#4 = F* = (UV)2 = (CT-i F)2 = 1*, of order 16;

(7.1) U*n = F3» = (UV)* = (J7-1 F)3 = (*72 F2)2 = 1,

of order 12TI3 ;

(7.2) U« = F6 = (?7F)2 = (U2 F2)2 = (17-* F)3* = If,

of order 96g3. And when m = 6:

(7.3) U2^ = F2*» = (J7F)3 = (?72 F2)p> = (*7F2)2*3 = 1,

of order 24p33 iip1 = p2 = 2, infinite for all greater values (so long asp3 > 1);

U« = F6 = (J7F)3 = (U-1 F)2 = (U-1 V-1 UV)2 = 1J,

of order 96;

(7.4) U5n = V5n = (UV)3 = {U-1 F)3 = {U-1 V-1 UV)2 = (*7F2)4 = 1,

of order 36(to5;(7.5) C76 = F 6 = (C7F)3= (C72F2)2= (C7F2)4= (C

of order 11520g5.When we try to extend these results to the general even value of m,

we are faced with the difficulty that, when j > 3 , the powers of S'Thave no elegant expression in terms of U and F. Thus (1.2) and (6.2)give no interesting results in general; but, by writing 2r for m in (1.2),and putting ps= oo for all j > 3, we see that the group

(7.6) U2pi = F2*1 = (UV)r = {U* F2)*" = (UV2)2*3 = 1

is infinite if r > 3 (and plt p2) p3 > 1).Moreover, in the case of (4.7) or (5.3), we can prove by induction

thatT8-* TS* =

This is clearly true when j==l. Assuming it for a particular j > 1,we have

= U-1. CH*-» F"1 UV*-1 .V=U-> V-1 UV*.

* This is a case of U*P = F1" = (UV)* = (U~l. F)» = 1,of order 4pq, which can be deduced from (1.5) by putting m — 2.

t Coxeter, 6, 72 (Theorem 6).| This is a special case of (7.1) or (7.2), with U replaced by U'1.


Thus (4.7) and (5.3) (with m — 2r) lead to the subgroups*

(7.7) U* = V2r={UVy={U-'V-1UVi)2 = l (j = 0, 1, ...,r-2),

of order 22r~1r, and

(7.8) u1**-*) = vn&-» =(uvy = (u-1 F)3 = (u-} v-1 uv*)* = 1

(j=l, 2, ...,r-l; r>l),of order ^n2r~1(2r)\.

The analogous subgroup of (5.4) is

(7.9) U«*-» = 7«(2r-i) = (jjyyr = (jj-i y-iy (jjyy = (jj-i 7)3

= l ( j = l , 2, . . . , r - l ;

of order ^2 r(2r)! . Here (£7F)r generates a cyclic central of order n,whose quotient group is (7.8).

When % = 1 , (7.8) is [S2*"-2]', the alternating group of degree 2r,generated by the permutations

U= Ka3a4...a2r), V— (a2a.iai...a2r).

8. Symmetry groups of the Cartesian frame.

The expression (5.7), suitably interpreted, is the general operator ofany of the groups (5.4), (6.3), (7 . 9). When m is even, (5.6) shows thatsx is of even " degree " in S, T, while t is of odd degree; also t is an oddpermutation of the suffixes. Hence (5.7) is the general operator of(7.9), if the symbol ( ) stands for any even permutation of the suffixes(and m = 2r). When m is odd, sx and t involve T an even number oftimes, and t is an even permutation of the suffixes. Hence the generaloperator of (6.3) is then (5.7) with the same meaning for ( ). Again,when m and n are even, sx and t involve T an odd number of times.Hence, in the general operator of (6.3) with m even, ( ) must stand foran even or odd permutation of the suffixes, according as SZt- is even or odd.

When n = 2, we can make these matters clearer by speaking ofpermutations of the Cartesian axes, and changes of sign of coordinates.The group of all such permutations and changes of sign, of order 2mm\,has been denoted by [3m~2, 4]. The following four self-conjugate sub-

* It is, perhaps, not quite obvious that these relations suffice to define the subgroups.The rigorous procedure consists in reconstructing the {£, 21} groups by adjoining to the{ U, V] groups the operator T which interchanges U and V. Cf. Coxeter, 4.

294 H. S; M. COXETER [Feb. 20,

groups (three of index 2, and one of index 4) are easily described:

[(3m~2)', 4], when only even permutations are admitted;

[3m-a, 4]', when the permutations are of the same parity as the numberof changes of sign;

[3m~3> *•1], when the number of changes of sign is even;

[3"1-3-1'1]', when both are even*.

The symbol [(3m~2)', 4] has not been used before, but seems natural,as implying that the symmetric subgroup [3m~2] of [3"1"2, 4] is replacedby its alternating subgroup [3m~2]'. The group [3m~2, 4]' consists of allpositive or rotational symmetries of the Cartesian frame. When m > 4,[3™1-3'1'1] and [3"1-3'1'1]' are the complete and rotational groups of thepolytope hym, whose vertices are ± ( 1 , 1, .-., 1) with an even (or odd)number of negative signs. Clearly, [3m-2, 4] contains all of [(3m-2)', 4],[3m~2, 4]', [3"1"3'1'1] as self-conjugate subgroups of index 2, while all ofthese contain [3"1-3'1'1]'.

The central inversion, which changes the sign of all the m coordinates,belongs to [3nir-2, 4] and [(3m~2)', 4] for all values of m, but to the otherthree groups only when m is even. Thus, when m is even, we have thefive central quotient groups

H3"1"2, 4], M(3m-2)', 4], H3m~2, 4]', K3"1-3'1'1], KS"1"3'1'1]',

which can be defined geometrically by agreeing that the simultaneouschange of sign of all coordinates shall be deemed to have no effect.E.g., when m = 4, we have

£[3,3, 4], M(3, 3)', 4], J[3, 3, 4]', H31'1'1], M31'1'1]',

which are Goursat'sf XLVII, XLI, XXVII, XLII, XXII, respectively.But, when m is odd, [3m~2, 4] contains the central inversion while its

subgroups [3OT~2, 4]' and [3™-3'1'1] do not. Hence

[3™-2, 4] ~ [3"1"2, 4]' x [ ] ~ [3™-3>!'!] X [ ]

• When m = 3, these five groups are:[3, 4], the extended octahedral group;[3', 4], the pyritohedral group;[3, 4]', the octahedral group;[3, 3], the extended tetrahedral group;[3, 3]', the tetrahedral group.

f Goursat, 1, 78, 67.


and

(8 .1) H3m^2> 4] — [3m-2, 4]' — [3m~3- *•*] (m odd).

For a similar reason,

[(3™-2)', 4]~[3"l-3-1 '1] /X[]and

(8.2) H(3m-2)', 4]~[3"1"3-1'1]' (m odd).

Thus there are only seven groups instead of ten; and of these seven only-four are abstractly distinct: [3"*-3'1'1], [S™-3'1'1]', and the direct productsof these with the group of order 2.

By the remarks at the beginning of this section, (5.8) is [3m~2, 4],(7.9) with n = 2 is [ (3 M ) ' , 4], and (6.8) is [(3"1"2)', 4] or [3"1-2, 4]'according as m is odd or even. Consequently, (5 . 9) is \\Zm~*, 4], (7 . 8)with n = 2 is Wtf2*-2)', 4], and (6.6) is [3™-3'1'1]' or H3"1"2. 4 ] ' accordingas m is odd or even. When m = 0 or 1 (mod 4), (6.9) and (6.7) providealternative forms for [3m~2, 4] and £[3m~2> 4]; and when m = S (mod 4),(6.7) provides an alternative form for [(3m-2)', 4]. By (8.1), we havenow accounted for all our geometrical groups save [32r~3'1*1], £[32r~3>1'1L(•32r-3,1,1-J'} a n d £[32r-3,1,1]'*.

I t seems worth while to give these abstract definitions explicitlyin the important case when m = 4 (or r = 2).

[3, 3, 4]: £8 = T2 = (ST)« = (S2 T)4 = (S-1 TSTf = (S* T)2=l.

[(3, 3)',4]f:

U« = V*= {UV)*= (U-1 F)3 = (U-1 V-1 UV)2 = {U-1 V-1)2 {UVf = 1.

[3, 3, 4] ' : R* = SS= {Br18f = {RS)* = (R2 S2)2 = R*Si=l.

^[3, 3, 4]: S* = T2 = (ST)* = {S2 T)4 = (S-1 TSTf = 1.

, 3)', 4]: *7« = F 6 = (UV)2= (U~1V)3= (U-1V~1UV)2=1.

, 3, 4] ' : Ri=Si

* [3"1"3'1*1] may be described as the subgroup of (5.8) whose operators involve S aneven number of times; it is therefore generated by <S'2, T, and <S-1 TS. [3"~3' l i l] / may bedescribed as the subgroup of (0.8) whose operators are of even " degree " in R, S; it istherefore generated by R2, S2, and RS. But no elegant definitions in terms of two generatorshave been found.

f Actually, this group has also the more elegant definition

S* = T2 = {S2T)3 = (S3T)1 = 1.


There are also the four groups [31-1'1], [31'1'1]', M31>1>1]> M31'1'1]'-But, of the whole set of ten groups, only nine are abstractly distinct.For, let iP1'1*1] be defined in the usual form*

O* = N2 = P2=Q2 = (PQ)2 = (QN)2 = (NP)2

= (ON)* = (OP)3 = [OQY = {ONPQf = 1,

and let us write R = NOP, S = OPQ.

It is easily verified that

RS = NPOQ (which is conjugate to ONPQ),

RJS-i = QNOQ (which is conjugate to ON),

{R*8*)*= [ONOQOPf= [N0PQf= 1,

whence

Q = R2 SRS-\ OP = SR2 SRS-\

N = SR2 JSRS-1 E~\ QN = R2 8R-1 SR8-1 Rr1,

0 = (R* S)2 R-1 SRS-1 R-K ON = (R2 S)2 RS~\

P=(R28)2RS-1Rj.

Thus H31'1'1] is generated by the operators R, S, which satisfy the aboveabstract definition for £[3, 3, 4]'. But these groups have the same order.Hence

(8.3) M S ^ ^ W ^ 1 ' 1 ' 1 ] .

In other words.. Goursat's groups XXVII and XLII are simply iso-morphic.

* This, without the final relation, is [31'11], by Coxeter, 1, 149 (16.74). The centralinversion is (NOPQ)3 or {ONPQ)3, by Coxeter, 2, 607 (iii) (n = 1).

| After some manipulation, these expressions can be simplified; in facts

N = R-*SRS* = S-lRSR\ 0 = SRS*R-\

P = RSR-*S* = R*S~*RS, Q = R^SRS-1 = S*RSR~K


All these nine groups occur in Dr. Burns's list of groups of degreeeight*.

Group.

[3, 3, 4]i[3, 3, 4][(3, 3)', 4]

[3'.M][3, 3, 4]'

£[3, 3, 4]' or ^[3*- »•»]K(3, 3)', 4]

[3L».i]'*[3i.i.i]'

Order.

38419219219219296969648

No. inDr. Burns's

list.

123456783

Her generators in terms of ours.

sx = S* T, s2 = TS-*s1 = ST, sz = S

nst = PON, s, = NPOQ

sx = R-\ s« = Sst = R, s2 = Sts1 = U, s2=V

sx = NO, s2 = OP, s9 = OQ

9. Groups involving the k-th power of the symmetric group.

The importance of the group (5.1) is doubtless connected with thefact that it is " only just infinite ", i.e., that its geometrical representationis neither hyperbolic nor spherical, but Euclidean (like the infiniteprogenitors of the familiar "groups of genus one§"). This property isshared by the more general group

(9.1) Smk=T2= {S~k T8k T)3 = (£-> TSf T)2 = 1

\; m>2)1T,

* Burns, 1, 207-209. There are so many misprints that it is advisable to compare herabstract definitions with their restatement on pp. 211-213.

f Dr. Burns's definition in this case is inconsistent with the given expressions for thegenerators as permutations.

J Here the relation (Sj2 sz)* — 1 is not an adequate substitute for (s~l 8t)3 = 1. Putting

pt = p2 = p3 = 2 in (7.3), we see that the group

U* = F4 = (UV)3 = (Ua F2)2 = (?7F2)4 = 1

is of order 192, not 96.§ Burnside, 1, 410.|| Meaning j= 1, 2 k — 1, k + 1, Jc + 2, ..., [frmk'].Tf We might have written

S"> = T2 = (8-kT8*T)* = {S-JT&T)* = 1 {j = 1, .... k-l, k+1, ..., [£TO]; m > 2*).

But the important case is when m is a multiple of k, since otherwise, on replacing S by asuitable multiple of itself, we should have (m, k) in place of k.

When m = 2, (9.1) reduces to (4.6) with p = 3, and so is not infinite but of order24.6*.


which is derived from the g.g.r. (3.2) with pk = S and every otherpf = 2, so that the representative graph consists of k separate m-gons.When we omit the last k of the mk generators Rlt R2, ..., Rmk {i.e., deleteone vertex and two adjacent sides from each of the m-gons), this g.g.r.yields the finite subgroup

(RiRi+k)*=l [i=l,...,(m-(9.2)

which is [3m~2]fc, the direct product of k symmetric groups of degree m.If we regard the i-th. symmetric group as operating on the m symbolsait ai+k, ..., a,-+(n,-i)*> we may write

By introducing the new operator

S= ( a ^ . . .

so that S-iRiS^

and

(9.3) ^•*=2?lJ21...

we increase the order from (m!)& to k{m\)k.On substituting S^TS* for J?,.*, (9.2) and (9.3) become

= (S~}TS>T)* = 1and

8k = JSH"*-1)* TS^m-1)k. £-(m-i)fc+i TSlmr'i)h~1... S~2 TS2. S-1 TS

i.e. Smk =

Since S is of period mfc, we can now exhibit our group of order k(m\)k as

* We could obtain the same relations by substituting S~'Tij' for i?,+i, and then weshould have T = Bt instead of

T = S^S-1 = SkRkS-k = /?•„_,,* ... R>kRiRik - . £(„.-!.*.


a factor group of (9.1), in the form

(9.4) 8mk =Ti= {ST}1*-1* = {8~k TSk T)3 = {8~f T8* T)2 = 1

( j = l , ..., &-1 , &+1, ..., B«n£]; m>2)* .

Thus (9.4) is generated by the permutations

8=(a1a2... amk), T = (amk ak)t

or, after a trivial change of notation, by

8 = K a 2 . . . amk), T =

E.g., when m = 4 and k = 2, it is the group of degree 8 and order 1152J.When (m— \)k is even, the operators 8 and T8~XT generate the

subgroup

(9.5)§ Emk = 8ml: = {R-1 S)Um-vk = {Rk 8kf =

of order \k(m\)k.The geometrical considerations of § 5 can be extended so as to give

an (m—1)^-dimensional representation for (9.1), in which [STfm-1)k

appears as a translation. The reflecting primes are nowxi = xk+i \? = 1, 2, ..., (m— 1)k],

and the number of dimensions is reduced from mk to (m— 1) k by putting

xi+xk+i+x2k+i+~>+x(m-Dk+i= ° (* = !» 2 . •••» ^ ) -

The longer translation (ST)n^m-1)k and its conjugates generate a sub-group of index T^"*-1)* in the subgroup generated by (tfT7)*"1"1^ and its

* For the rest of the paper, we shall forgo repeating this statement of the range of j andthe restriction on m.

A consideration of simple cases seems to indicate that the relation (S~kTSltT)3 = 1 isalways superfluous (as we know it is when k = 1).

j It is natural to expect that the analogous permutations

(axaa ... a,,,), {a^k+i),

where now m is prime to k, will generate the symmetric group of degree m in the form

S"> = T* = (STY'"-™ = {S-kTSkT)3 = (S-'TS'T)* = 1 (j = 1, .... k-1, k + 1, ..., [Jm]).

The sufficiency of these relations has not been proved.I Burns, 1, 210.§ When m = 4 and k = 2, this is Dr. Burns's " Order. 576, No. 3 " (with a different

abstract definition). Actually, this group is the same as


conjugates. Thus the quotient group

(9.6) 8"* = T* = (8T)ntm-vk = {8~k T8k T)8 = {S-* T8* T)2 = 1

is of order lcdm-^k{m !)*.This is derivable from the larger group

(9.7) 8nmk — T2= (8T)n^m-1)k = 8~mk T8mk T = (8~k T8k T)3

as the quotient group of the cyclic subgroup (central) generated by 8mk.The group (9.7) is thus of order kn^m-^k+1(m !)fc. I t is easily seen tobe representable as a permutation group of degree nmk in the form

8=(l1l2...lmk2122...2mk...nln2...nmk),

T=(l1h+1)(212k+1)...(n1nk+1).

I t is a subgroup of index w*-1 in the group whose general operator is

where ( ) denotes a permutation of the suffixes in accordance with thegroup (9.4), and s ,= (l,-2t....?*,•). The index is determined by the factthat the only combinations of the s's alone that belong to (9.7) are theproducts of conjugates of

s1s2...sk =

i.e., those combinations for which

+"- = '-' = h+kk+"'+imk (mod n)-The operators ( ) that can occur unaccompanied by s's form a subgroupof index k in (9.4), namely the subgroup generated by T and its con-jugates. I t seems probable that the whole group of order knmk(m\)k

cannot be generated by two operators (when k > 1 and n> 1). E.g.,when n = m = k = 2, this is the group of degree 8 and order 128*.

As analogous generalizations of (6.1) and (6. 2), we have

(9.8) Rmk = 8mk = {R-18)Wm-*>k = {Rk Sk)* = (& &)2 = 1,

of order %kn{m-1)k(m\)k, and

(9.9) 8mk =T2= {8-1 TST)Wm-v>k = {Sk T)6 = (8* T)4 = 1,

of order hdm-^k{m \)k.

* Bums, 1, 209.


References.

J. E. Burns, 1, " Abstract definitions of groups of degree eight ", American J. of Math.,37 (1915), 195-214.

W. Burnside, 1, Theory of groups of finite order (Cambridge, 1911), 419.H. S..M. Coxeter, 1, " The polytopes with regular-prismatic vertex figures", Proc. London

Math. Soc. (2), 34 (1932), 126-189., 2 , "Discrete groups generated by reflections", Annals of Math., 35 (1934),

688-621.-, 3, " The complete enumeration of finite groups of the form B,-1 = (RjRj)ky = 1",

Journal London Math. Soc., 10 (1935), 21-25.-, 4, " An abstract definition for the alternating group in terms of two generators ",

Journal London Math. Soc, 11 (1936).6, "The groups determined by the relations S' = T'" = (S-1 T^STy = 1", Duke

Math. Journal, 2 (1936), 61-73.H. S. M. Coxeter and J. A. Todd, 1, " Abstract definitions for the symmetry groups of the

regular polytopes in terms of two generators", Proc Camb. Phil. Soc, 32 (1936).E. Goursat, 1, " Sur les substitutions orthogonales et les divisions r£gulieres de l'espace ",

Ann. Sci. de VEcole Norm. Sup. (3), 6 (1889), 9-102.G. A. Miller, H. F. Blichfeldt, and L. E. Dickson, 1, Theory and applications of finite groups

(New York, 1916).E. H. Moore, 1, " Concerning the abstract groups of order &! and £fc! holohedrically iso-

morphic with the symmetric and alternating substitution-groups on k letters ", Proc.London Math. Soc (1), 28 (1897), 357-366.

J. de Seguier, 1, " Sur les Equations de certains groupes ", Journal de Math. (5), 8 (1902),253-308.

A. Sinkov, 1, "The groups determined by the relations S'^T"1 = (S-1T~1ST)»= 1 ",Duke Math. Journal, 2 (1936), 74-83.

A. Young, 1, "On quantitative substitutional analysis", Proc London Math. Soc (2),31 (1930), 273-288.

Trinity College,Cambridge.

Documents

The Abstract Groups Rm = Sm = (Ri Si)Pi = 1, Sm = T2 = (Si T)2Pi = 1, AND Sm = T2 = (S-i T Si T)Pi = 1