PROCEEDINGS OF A SYMPOSIUM IN PURE MATHEMATICS
OF THE AMERICAN MATHEMATICAL SOCIETY
Held in New York April23-24, 1959
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PROCEEDINGS OF A SYMPOSIUM IN PURE MATHEMATICS
OF THE AMERICAN MATHEMATICAL SOCIETY
Held in New York April23-24, 1959
Cosponsored by
THE INSTITUTE FOR DEFENSE ANALYSIS under contract Nonr 2631(00) with the Office of Naval Research
iinitorial Committee A. A. Albert Irving Kaplansky
696I ONV'ISI SCIOHU
.SCNSCIIAOTId
AJSICOS TVCIJVIAISHJYI
I
NVCIT{SWY
sdnouo SIINI.{
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SCIJVIAIgHIVI{
gUNd NI VISOdIAIAS
do scNrcrsscotrd
Prepared by the American Mathematical Society under Contract Nonr 2631(00) with the Ofrce of Naval Reeearch
International Standard Serial Number 0082-0717 International Standard Book Number 0-8218-1401-X Library of Congress Cataiog Number 50-1183 Copyright O 1959 by the American Mathematical Society Third printing,1979 Printed in the United States of America reserved except those granted to the United States Government May not be reproduced in any form without permission of the publishers
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contained in
II. It then foll.ows that N*(H) S NX(V). If we further
suppose
P, is a p-Sylow subgroup of Nr(H), then unless Pt is a p-Sylow subgroup of X, it follows that Nr(H) < NX(V), since N*(Pt) S N*(V) and
that
NX(P1)
{
w*(H). lntuitively, playing off Theorem B against suitably cho-
sen subgroups of P1, there is a possibility of finding p-subgroups of X whose normalizers have prescribed properties and which involve the
prime p to a considerable amount. This explains the underlying idea of the proof, which otherwise becomes obscured by details.
It might
be well to mention some ideas which don't seem to work,
though superficially they seem relevant. The
first
of these concerns the
maximal p-Sylow intersections P OPx of a group X. One might think that a knowledge of these groups, with
their normalizers, would lead to very
precise information regarding the way in which P is embedded in X. This may in fact be the case, but these groups seem very difficult to manage, and my impression
is that knowledge of them provides only a crude ap-
proximation to what is really going on. A second idea would be to keep an eye on the terms of the ascending (descending) central series of P, and to study
their normalizers. This idea broke down, apparently because
if A is a term in the ascending (descending) central series of P, and A . Pl < P, there is no reason why A should be
of the following possibility:
a
term in the ascending (descending) central series of Pt. ln any case, after considerable floundering in the central series, it
occurred to me that as regards Theorem A, the relevant p-groups are the maximal abelian normal subgroups of P. Here, abelian normal subgroup of P, and mal abelian normal subgroup of
if A is a maximal
A ' Pl < P, then A is also a maxi-
Pt.
Even more fortunate, however, is
It is this last property which, at a very critical point in the proof, makes it possible to assert that a parthat A is a p-Sylow subgroup ol Ct(A).
ticular p-element, about which information is sorely needed, actually lies in A.
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BURNSIDE GROUPS AND ENGEL RINGS by R. C. LYndon
$1. Introduction
If G is a discrete group with descending central series Gl Gn+1 = (Gn, G), then the formation of commutators (u,v) =
"-1
= G,
u-1
rru
L on the direct sum of the abelian quotient groups Gn/Gn*l. If G has prime exponent P, uP = 1 for all u in G, then L has characteristic P, PU = 0 for all U in L, and moreover L induces the structure of a Lie ring
satisfies the (p-1)-st Engel condition:
(E-,): 'p-r'
[u,vn-r1=tt... [u,v],...1,v]=o
for all u and v in L. we are concerned with the question of whether, in general, L is subject to any further identities that are not consequences of characteristic p together with the Engel condition. Investigations by Sanov and Kostri.kin strongly indicate that there are no such further iden-
tities, but are not fully conclusive. we approach this problem by a method introduced by Grirn, which we are able to simplify for the purposes of the present context. Although we are far from proving that the identities
of Grii'n are consequences of the Engel condition, or even of recovering the results of sanov and Kostrikin, we think it is worthwhile to present the problem in a direct and elementary form.*
The connection between a discrete group G and a Lie ring L was
first
investigated by Magnus, and by
witt,
Zassenhaus and others, nota-
bly, in a recent paper, by Lazard. This connection has been fundamental to much work on the problem of Burnside. Recently Kostrikin has shown
'I
error, due to faulty arithme1957, p. 309.
take this occasion to acknowledge an
tic, which was pointed out by Kostrikin,
I
u> edlu (u)C,d11rc11dxa Ir fluo puE ll 0 = ruV U - rV l€rn s/t\oIIoJ lI' 6uCu = e ,(u+rn)" > ((")c ,(*)c) ," (r)" :(du)c, =
a(t)c
qclq^t roI so-rras Surpueasep fipldz.r lsoru aql sI slq] Jol serJes uolsuaulp-d aq1 yo (u)p rrrral ql-u aq1 o1 s8uolaq n Jr
:c
pu" JI
ul sl I-n'.ra.rroa.roy41 'C
^Iuo
q
n oluos ruorl snql alq€urElqo si I-A ,A aql u! = A arrros ur u aat8ap ;o snoaua8ouroq lualuale all-d d.rarre pue
uv
luatuala
all-d
U pu€ 'C
ug
I
u6t
olnpotu luan.rSuoa s1 s,ternlr lnq lr*uv u-r s-r 1+uv '1-a = A urelrac ur u aa.rSep ;o snoaua3owoq sl ug a"aqrn
e
01
u+un+I=o+I=n urJoJ aql ur suorssa.rdxa duetu arreq dlr.reugpJo ll-lltr
C Jo luauala uV 'dA sramod ql-d puB A/t\ uoll€ruroJ raprm pasola pue O
:n aql
'I
- tl1
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V Jo elnpotnqns lsolprrrs aql
dq palz.rauaS V Jo (8ur.r ar1 pa1cl4sa.r) Sur.rqns
all-d
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-r?qc
Jo
X pIeIJ aw1.rd eql ra^o p dno.r8 e;o s8u1.r ag1 pue sdno.r8
3u1.r dno.r8 aq1 aq
V laT
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?
aqI
ru IIE roJ 'ru luauodxa
qlr^\ pue s.role.raua8 o,rl uo g dno.r8 allulyul uB Jo acualslxa aql
3ugqs11
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a8eul alqd.roruornoq
"
sl
.Z$
''D/C
=
p
lBrll
^o>lr^oN dno.r8 allqJ ar{l
sJol€.raua8 yo raqunu.re1ea.r8 ou uo d luau
up e*o" qlr&\ llrelsuoc sawoceq C Jo salJes I"rJueJ Sulpuacsap aql uaql 'd luauodxa Jo pu" pele.raueS d1e11u
-odxe ;o dno.rE a1ru1; d.raaa puz I
'C
=
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1zq1
G(.) =nG1n) = 1. In any case G = G/G1.; is faithfully represented in T. A/ t' , with the induced map of G into f, = : Ln/Ln+l mapping =
Glrr;/G1rr*r; homomorphically onto
1n
I a'\+L and
car rying commutation
into commutation and p-th powers into p-th powers'
If w = uv, writing w = 1+W, u = 1+U, v = 1+V, from W=w
-
1 = uv
-
1 = (1+U)(1+V)-1 = U + V + UV
it follows that A is in fact generated by the elements X = x-1 and X' = x-l -1 for any set of generators x for G; if G is a torsion group n, any n, it suffices to restrict the X' are superfluous. (Indeed , f.or A/ A the x to a set of coset representatives of a set of generators for G/G2.) In general, fromO = X + X' + XX', iterating the substitution X' = -X - )O(' shows that
X' agrees modulo An, for any n, with the n-th partial sum of
the series (1
+x)-1
-1=-X+x2-
which leads to a representation of G in A, ti
"
obtained by completing E according to powers shown by Magnus and Witt that
if
Lu
= 1,
= 0. whence
G(r)3G(r)
We are concerned here with
nent p: G =
r/FP,
power series ring in the X
of ,l . By this
means
it was
G is a free group, on generators x, then
andthatL is free
on generatorsX'
a'relatively free' group of prime expo-
F free, FP the normal subgroup generated by
all p-th
powers uP of elements u from F. Since O(n)O = 1, the dimension series reduces to the descending central series: G(n) = Grr. Since the quotients
of the descending central series have exponent p, L has characteristic p' and we may as well take K to be the prime
of Novikov, we will not in general have
G,
iield Zn. In view of the result = 1,
nor a'
0; but, since
G is a torsion group, we have no occasion to introduce power series, and
it is optional whether we reduce A modulo ao . In any case' if G is relatively free on generators x, it is easy to see that each ull in G = G/G .
L
"k..'tk("t, "''lt)e'-
I, u {nt* "' 'x
"[,.
{!t"'tII
3
r
,I[d1 >_s]d I 1t{tI
*Ij
r
)
=
r - 6 {trx*r)''' (Ix*r,} uaql,n+
1 = O(lx
=n
. . . Ix)
=
nU
'A\ou suollsanb asaql olul raluo 01 aJnleurard rnaes
'x
aql roJ sluatnala ar1 d.re.r1rq.r" ]o uorlnlllsqns ,{q slueluala ar.rauaS qcns ruor} peulslqo aq u?c 'd6v olnporn lsEal 1€ 'd Jo IIe l€ql /t\oqs plno/rl 1I lnq
plnol\ ,sseco.rd Eurualq8!?Jls, ? l€ql uollecjpur euros sr a"raql .lcullslp tt*, .,,21x,t," tt* . .' tt*tt* n tuo.r; Buysl.re,6 = A roJ
""r
uA =
"""u^
Jo slueurala ,c1.raua8, dluo .raprsuoc
an 'uorlecryrldruls JaqUnJ E sV
'6,r = n rrrrol aql Jo sluarnala ;o
u61
s*"a1 Bulpeal or€ l"ql d
s.role.raue8 asoql dluo raplsuo? acuaq puu 'd6 ueql ssal aa.r8ap ]o o1 uo11e311saau1 luasa.rd aql
.ro; ,da1s
lcrrlsar an flrcrldtnrs
'(6rv oFPo*) alrt3
=
I - (6la* I)r
=
Jo
swrel
lsrg
E sV
I-n= o lBI{}
lrr' "Zt'In'dln'''dZndln a,r"q eirr'UA + I = 6(rt + t) = A^ aculs'J ul un = n rrrro] aql seq 6J Jo luernala le.raua8 a{I .d.{ ul n sluawela Jo stnral Surpeal 1e dq pale.raua5 1 ur IBapI aql sl d erar{^\ d./.1 ='I ,{11uapr -lg'arur"rde s1 d a.raqm'6Jx = x sroleraua8 uo dd/d = 5 dno.r8 aplsurng aq1 uroJJ
palrrap
I - x = X sroleraua8
uo d crlsrJalc?Jeqc;o Bur.r ar1
aql aq T la.I 'x stole.raua8 uo g dno.r8 aary (d1a1n1osqe) eq1 uror; palrrap
I -x=x
srolsraua8 uo d a-rlslralcBrEqc ;o 5ur-r olT oarJ oql aq T la-I
unrc
.I
+ uc ur uc ul lou lnq
Jo
salMuapl aqJ .gs
n ,paepur ,pu€:1+ur ul sl u pu€ 'u auros JoJ u aa:3ap Jo luaiuala ar1 snoauaSouroq E sl 0 + u1 e;aqn
,l
'u*tn+I=o+I=n 'I-x = X aql ur lenuoudlod e s" uolleluasa.rda.r
anblrm e seq
where 0(iL, .. . , js) is evidently the number of ways that the sequence o = (jf . . . j") can be written as a product of p non-empty strictly increasing sequences o(1), . . . , o(p). More explicitly, let a = &(i1...is)
in c, j, . ii*1. Then d is the number ofways,afterinsertingabreakateachofthe F =s -1 - a non-increasing steps, of choosing the remaining b-1) - (s-1- c ) = p-s + o breaks
be the number of increasing steps
amongthe a increasingsteps: a =(ois1 u=
1+
=("10). tttns,
") ("(:t "' j"))x
: t pssstp j1,...,j"\
s-P- /\,"'1"'
If F is any polynomial in a set T of indeterminates X, which need not commute, then F = : . _ Fsu where Ft is the sum of those terms
^ o<s
that contain precisely the indeterminates in the set S, while inversely,
> " o
Fa
=
(-l)cardinal (s-R) r(n), where F(R) is the result of sub-
stituting, in F, O for all X not in R. It follows that, instead of the generic u = 1 + U considered above, we may consider the 1 + U*, where U* con-
sists of alt those terms from U that contain all of X1, . . . ,
\.
The lead-
ing term of such a U* is
st = xt "[:o)l
*o,
. . . x'.,, t ZP,
where the summation is over all permutations
Proposition
1. For all t, P ( t
< 2p,
o of the indices
1,2, . . . ,t.
E satisfies the identical relation
st(Ul, .' ., Ut) =0, all U1, . . ., U,inE. This was established for U1 = X1, . .
'relatively free' on Xl, . .
.,
Ut = X,, and E is evidently
.,Xt, . . . . More explicitly,
choosing u*
=
1+ U* modulo ot + 1 and u, with leading term Ui, the substitution of u, for x, in u* evidently yields an expression with leading term
st(ul,...,ut). Let ,1," denote left multiplication by Y, and rt denote right multiplication; then the inner derivation dt: X + [X,y] = XY - YX is given Uy dy =
6
'a8ue.r s-rql
ul 'acuaqn 'd >
rI
qlln fpo pauJacuoc aJ? alrt ,dg >
1 .rog
'V Bur.r aqt ut
O=(ln""'In){'lS ,"I .r.l ln,...,Inpue,4+d ? lIIErod .arnlcoluoC aq1 slsaE3ns srql
1^
I
"ox'"'lxt111,;r
={'1g
uolllulJap le.raua8 arotu aql Sulryt{
':f 3ut-r aqt ut o=
'f
l'n'''ton tT
a
= (1o
"''
'Io)o'ls
ln ' ' ' ' 'Ifl 'd ? I ile "o.{
'Z uor.llsodord
-I tuo.r1 In "ql "oy saqsluel 'JIaslI T ol ueql raq}"r
'!x
sluaurala Jo uorlnlrlsqns raprm.{11ecr1uepr
y 8ur.r Surdola^ua aql o1 s8uolaq le.reua8 ul qclq^\ ,O'1g luatuala aq1 I *l > I > I + 'd ? 1II"ro''acuaq,tt'I"1I k'''''l*'''''IX)0'1S 0'I + 1g d1.rea1c,1' x' . . to" K = o'1, Burlr.r,tr,lce] uI =
T
o=
Jo
ds
^fir1uapr
'uorlrpuoc 1a5ug aq1 ...
aql puB'[1-a{'x] ;(1-d)rsan13 u, u,
dS^
=
\
=
=
sa11dru1
k
= zx
= 'X' '' ''* a = r"ql ra}ur 0 = [1_AI'X1 uolt-rpuoc la8ug eq1 uror; pu" 'dx ' ' ' ' 'Zx o1 lcadsa.r qlrrn ,{laarssacans ozrr"aull deru an'I-d' 'I'0 = l'r'.reln8uls-uou s1 (r1) apuoruJapue1 aqlaculs'I-d' 'I'0=l\ 'dXdY * "'*ZXZ\ =I'IX=XBuJllas Eu111as',{1as.ranuo3'0
r-d=r;0 g-1-6trX1l 3 r-r-dtr" r^ r (r1d)r(r-)
r-d
t I t o*
=
=
1-6(Ia - ^")* = r-olP.x = [r-dl'x] 'd atlsr.ralcereqa seq y pw 'e1nw*oc tr" pua LT aculg 'tr T - L,
the conjecture is equivalent to
>o(o)kuor. We pause momentarily
.
Uor=0, allp+k ( t < 2p.
for an historical digression. The method of
Griln, restricted to the present conte:rt, is better paraphrased in a slightly different fashion. Embed B = F/FP, relativelv free on generators x, in B'
relatively free on the x together with an additional 'generic' generator y.
If C is the normal subgroup of B' generated by all conjugates of y,
then
B = B'/C naturally, and Co = C/ (C,C) becomes a cyclic B-module. Writ-
ing cO = c(C,C), the coset in CO of c in C, one has in particular that
.o'X1 . .. Xt= (c,x1,.. .,
*t)0.
Grirn exploits this relation by substituting
for y elements from various
compound commutator subgroups of B. Under this representation of on the module CO, one has the additional relation Xp-1
of our relation St = St,t-p = 0 for small
-
E
0, and instead
t Z p, one has the relations
St,t-p*1 = 0. If B is infinitely generated, the restriction of this representation to
f, annihilates at most the centralizer
of.
f,
in tr.
S4. Relations among the identities of Griicr We conclude with some fragmentary results that exemplify such methods as we have found
Proposition
3.
For
allt Z p + 1, and U1, . . .,
st,1(U1,,.., in the ring
for investigating the conjecture stated
Ut)
above.
U, in L,
= !o(.)u al,..uo,=o
f.
We shall give two pro<-rfs of this proposition, which have different
generalizations. Our
first proof depends
on the following
10
II
't' * "' to"
t1,Jo
)s = (Ix'""u";{'1, leql alou
Sursea.reap Jo raqumu eql
'(, )" -I-t
=
(t)d
o,r\
, < Io
(I+l
sdals
8ur1r.ur ure8e acug
.@ tI4aIIrur{F € si >i'ls [aqt-]Tlleer.tuapl g = I-)i'I-it
II
.9 uor.llsodord 'SrrrwaI
aql Jo acuanbasuoc InJasn arour E u-relqo
aA\
srallal luacefpe Sursodsuerl
dlaarssaccns dq paurelqo aq uBc uorlelnru.rad ,{"ra,ra
.(1x,...,I+lx,*rx,
_ru* snoeua8ornoq
1eq1 1cB}
aql iuory
...,I, x,Ix,...,Ix)* {'ur+ls, e 3 _ (lx '" ''I+lx'iA'lx ( " "Ix)I+{(I+ls
'rlfeT5failffi'rua@ " X " c a = A\ loT .? uorJ.sodord E aq " X "' I
'uorlecrldde ou
prmoJ
o^€r{ eA\ qalq^\ roJ '11nsa.r 3urino11o1 aq1 saa13 Brnural s-rql Io uorl€rall
(n\-('\=rr-{\ \l - (")"/ \r - t')" \t"1"
/
)
olNIxr+lxH= oy.ro;pue
,( n \ -('\ =/'-n \ \r - t"1"7 \1.)"/ \r - (.)"/ uorlEI
-ar aro 01 sacnpor sluercrJlaoc
-
aq1 yo,{111enba
'yl+l*l*t
to =
t'* t ...
Jod 'uorlenbe aql Jo Joqwaur Jaqlla ur lualclJJaoc oJaz-uou 'y qlrAr rncco lxl+lx ro I+lxlx 1.red e qlr^{ suJel dluo leql lsrl} aloN
'(k'"
ru.re1 B
'
'!x'I+lx " "'Ix){'lS - (1x" " 'I+lx'lx '.. .,Ix){'ls =
(lX,..',Jl+l*rl*],'..,IX)I-{'I-lS '€uIuraT
From Proposition 5 it follows that
Ir"\j-t /t-t-"\llf sJr k /) l\x'-\
x^o1 ...x ot
=0.
- (t-i-") is a polynomial with leading term < k,weinferthat Io hx-c =0forall 2 ,k/xl. If S..=0forallh t,h h < k,and,from :f(")X o =0,that2 lnkx., =0. SupposingP > 2, kx we have also that : o o = 0, whence St,k = 0. This establishes If k is odd, f (o ) = (;)
6. Ifk is odd, St-1,k-1 = 0, St,0='..= St,k-l
Proposition
p > 2, !irg! St,k
= 0, and
= 0.
This suggests trying to establish by induction on k the condition Ck' St,k = 0 for
C, ,
allt I
p +k. Proposition 5 then shows that Cg, . .
.,
implies C,- provided that k is odd. Since Proposition 2 gives C^,
^-l-s
from Proposition 5 we infer Ct. Our second prooi of Proposition 3, that is, of C1, can be generalized, at the price of some complication, to establish C2, whence, by Proposition 5, C, follows; but the complications seem to prohibit an extension beyond
this point. Summing a set of substitution instances of St-1,0 = 0 gives
o=
Xo
n?
o
t,,o("l,..., *h,..., *k,...,
[xn,xuJ) = o.
coef o.t '., x ot - arises once in the expansion of A with ficient +1 for each part *or* or*, *t,n o, . oia1, an4 once with coefficient -1 for each part X oiX oi*l with c, > oi+1. Therefore the total coefficientof Xo inAis "-p= " -(t-1-")=2 a -(t-1),andA=
A term
2S,,1
=X
- (t-1)St,0.
Supposing again that
P > 2, it follows that S,,1 = 0'
To establish C, we begin with the identity
A=
r St-2,o(Xl,... , *i,..., *r,... ' *p, '..,
ik,.. . xt, [xixj], [xnxol) t2
=
o,
TI
ul€lqo aa 0 = O'Z-]q urord ',O- = O-
=
=,q+,J,0 =,CI* O,0 = q +,C suorlelaJ,(*) qllrn Z tq1 ur'urelqo a^r., yo1q31.rpu"lJaI (f ...1) * (l ...I)
,C = C acuaq^l'0
d.rlauurds
I€sralor aq18urd1ddy.0 =C+ C a^€qa^\puy',e _ CI _ ,C+ C = zv u"qJ'[zre] * [slz] =,c'[rsz] + [zsr] = c'[rzt] =,c,[tzr] = c tar 'o = [i8z] + [zsr] + [ezr]
(,r)
'snqJ
.[ozr] = txlx(lx,.'.,tT,...,I",.'.,Ix)o'z-t, aAEq a^\ g =
'[ozr] +[rez] +[zer] '1eu1ru.re1
'asr^\roqro 0 pue
'ox
t
t
i
=o
o'z-1, ruo.r;,3 + d !
*[szl] = ox,i
1 acurg
-I'1r=s
sr .ro a1dr.r1 e sur3aq raqlta .rred Bursea.rJul r{cea acurs
lo I l-ln
Jr
I = {lort}
"""u^
'x
{,tozt} .
=
[ozr] rer '[rrz] - [zre] - [rez] - [zrr] - [rze] + [rzr]
zv u"qr = ,lrB o ,E o
s saldr-r1 a^nnJosuoD Jo req.,,nu aql aq {'ntr} t"t'e'z'r;o uorlelnru.rad due 4'f'1 rod 'I+" = q qll/r\ ,s! leql ,1u1ots1p lou are leql q > qll^\ (t+q,q) " '(1+e'e) s.rled uro.r; uorlnqrrluoc aql sI ZV aJeq/t\ ZV * V = Iy .1ae; u1 {./rfr} :
= [:rtr]
lar
'>{'['! ot crqd.rowosr .rep.ro Z+€
.g
o*rr
= z,lg ecuaq
3 aa13pFo.{\0=Iyacuaq.Tr'D ulcllerpenb'3 < pu€0= roJ'srt4crr4t^'Z/.fi - t- rd=Z/Z(d - ")--Z/ed + tl"Zsureluoc luarcrJJaoc aql Jo r.ural crle.rpunb oel
:'x { t!u'I + do - t9l } I I
=-
Iy
a.,\r8 plno^r
, o _I_l = €/
-
I+Qo
)u8s .1Bo
Jo rrrns aql eroJaror{l
r"y
acws
stql'uorlrpuor ssaurrurl
-slp eql ruo.r;1redy'lcurlslp ar" I+q'q ,f+e,e ler{l qcns ,q > 1Qr,
d
ra^o " IIe
- I+" o )u3s
sI v uI
'x
Jo luarcrJ;aoc
aqI'q > r pue'rl > q'[ > I lsql qans >1,q,[,1 lau]lsp IIB ralo pauruns
o=,.i.0t,-r,o (X1,''', ir''' " *i''' " lk'
.
', Xt. [x,x,xoJ) = [rzs] - [zrs] - [stz] + [szr]
=C-D'+C' = 3C. Supposing
p>3, we conclude that C = C'
= 0, D =
D' = 0' hence A, = 0'
SinceA=0,itlollowsthatAr=A+A2=0,whichcompletestheproof'
?. If t Z p +2, andP > 3, litelSt,z = 0' Corollary. Ut >p+3,andP > 3,!hg!St,g=0'
Proposition
UniversitY of Michigan
BIBLIOGRAPHY
W.Burnside,onanunsettledquestioniq.thetheoryofdiscontinuous "' (1e02) 230'
?ilp;;b"i.i.
o. Griin.
J' Math.
35
Zusammenhang zwischen Potenzbildung und Kommutatorbildung'
Crelle 182 (1940) 158. between periodic groups and Lie rings, A' I. Kostrikin, on the relation (1957) 289' 21 mat. Izvestia, ser'
A.I.Kostrikin,OntheBurnsideproblem,Izvestia,ser'mat'23(1959)3' M.Lazard.SurlesgroupesnilpotentsetlesanneauxdeLie,Ann.Ecole ---
\or-.'sup. (3) 71 (1954) 101. Magnus, Beziehungen zwischen Glgppen und Idealen W. "' -;;Ai;ilen Ring, ilIath. Ann. 111 (1935) 259'
in einem
Crelle W. Magnus, Uber Gruppen und zugeordnete Liesche Ringe'
182
(1940) 142.
P. S. Novikov, On periodic groups, Doklady Akad' Nauk
SSSR 12? (1959)
149.
I.N.Sanov,Theconnectionbetweenperiodicgroupswithprimeperiod ^' ^";"d"Lib i;;;;ti;' ser' mat' 16 (le52) 23' "ing", 1?7 (193?) 152' E. Witt, Treue Darstellung Liescher Ringe' Crelle endlichen p-Gruppe' einem LieH. Zassenhaus' Ein Verfahren,..jeder Univ' Rine mit Oer Crraiitleristiti p zuzuordnen' Abh' Math' Sem' (1940) 200. 13 Hadburg t4
9I IIB ruorJ lurolsrp sl V
-VgV luapuedapul
Jo
ltq] u^\oqs
d11sea
sl ll C dno.r8 e qans uI .sdnor8
arnlcnrls alalduroc aql paururralap s"q roqln" aql
Jo IEuJnof u"rp"u"C aql ur par?adde qcrqrrr Jad€d € rq ='scrlerneqlery v
'uralsraH qlrirr dpulot paul€lqo ora^\ srar{lo pue .{lluapuadapur paurelqo ara/h sllnsor asar{l
atuos 'uo 1.rod3r 01 qslt\
I
qclq/r\
{ro^\ slql
s-l
ll pu" 'sdnot8-ygv
Jo
ul lsa
-ralur rorprnJ Io I"ap poo8 e palelnw-rls s€q >lro^\ d.reurturla.rd s1q;,
'g
1o dno.r8qns IBurIxEur B
sMeql
qcns Jo arnlcnrls aql paururralap pue
uorldrunsse aql raplm C sdnorg
'p ;o sdno.r8qns are g
pue V araq^\
'g ul I t' q'v uI ,e'€ ',€q? tu.lo; aql Jo uoll"luasarda.r anbrrm € aAEq ro v ur raqlra ar" qctq/t\ Jo sluawala oql 'c sdnot8 'sr 1eq1-sdno.r5-ygy tuapumput Jo arnlenrls arrres aq] 1y'c11c.{c
ar" g
aq1 pa1e811salur
roqlne arn pu€ rarrruaZ arnll
sl dnorg VgV u" 1eq1 e.rnlcaf g pue V qllrrr ,gVgV inroJ aql Io
pu? V Jt alq"^Ios
-uoc ol uralsraH pal {ro^t s1q; 'c11ada
uo-rlzluasa.rdar € llrup" plp lnq .VgV ruroJ aql Jo lou sezn (d,g)TSd l€ql pa^\oqs osle .,t4sue1dey
g
puB
'rapro arurtd
Jo
sl
auo lseal 18 pue arladc a.re
V JI alqe^Ios s! dnor8 E qcns leql pa./(oqs uralsraH pue d4sueldey
'VgV rrrroJ
aq1 yo sdno"r8
;o a.rnlcn-r1s aq1 Sulu.raeuoc paultlqo ara^\ sllns
-a;1e11red awos acuaraJuoc aql 3u1.rnq'lg6I jo rerrrurns aq1 ug e3a11o3 urop/t\og 13 plaq sdnorc ;o d;oaq1 aql uo aJuaraluoc un8aq a.rern uollearlp slql
q
acroJ rtv u? 13
suorleSrlseaul owos '€I pue
y
sdno.r8qns ;o
sIrIJal ur uorl€zrJolce; palecrlduroc aJorrr B lrtupt qcrq,rn sdno.r$ uo auop uaaq p€q >lro.r
anlll ,{.ral ,{11uaaa.r ,{.ral l1lun 'gV
rrrro} aql 1o sdno.r8
Jo oJnlcn.rls aql lnoq€ u,roro{ sr lunorue olq€raplsuoc e q3noqlly
uIalsuaJoC 1a1utq ,{q
sdoouc sTsv^Tos Nrvrusc do sunJcouJ,s
UHJ, No
group' If M is its regits conjugates, and consequently G is a Frobenius h' then O(G) = hm and h 1 m - 1' ular subgroup and O(M) = m, while o(A) = et'l I hm' This on G imply in addition that (1 . Our special assumptions
M of a Frobenius group is additional condition on the regular subgroup abelian group of type (p'p' sufficient to imply that M is an elementary
nei-
(with two possible exceptions . . . . , p) and that A is maximal in G ABA-groups)' In parther ol which can arise in the case of independent is solvable' The argument is alticuLar, it lollows that G is solvable if A
require the fact that the regular most entirely arithmetical, but it does (but only in the case when subgroup M of a Frobenius group is nilpotent M is known to be solvable)' is the group G of oneThe prototype of an independent ABA-group field F' If a affine transformations xt = cx + d over a finite
dimensional
denotes the transformation
x'
= wX, where w
is a primitive element of F'
andbthetransformationx,=-x+l,thenGisanindependentABA-group B (b). In this case the reguwith respect to the subgroups e = (a) and = group over F' This examlar subgroup M of G is simply the translation plealsoServestoindicatethekindofstructuretheoremswhichholdfor cyclic' ABA-groups in the case that A and B are non-independent
Herstein and In a paper in the Canadian Journal of Mathematics'6 G in which A and B are cyclic the author have shown that an ABA-group
ofrelativelyprimeorderisnecessarilysolvable'Itisnotdifficultto group always contains a normal subdeduce from our results that such a
n T = 1' where T is either group T such that C = N(A)' T withN(A) group of odd order and the abelian or the direct product of an abelian of G are either quaternion group. In particular' the Sylow subgroups group' abelian or isomorphic to the quaternion and the theorem of Grnn.13 The main tool in the proof is the transfer transfer of G into A maps G onto A; When N(A) = A' it is shown that the 16
LI
-VgV
roJ uorl"zrrolc€J ^ue
"
Surrunsse 'swalqo.rd aseql Jo lsrlJ aql
learl
01
u" leq1 ,{1a>1r1 ^{.rea pawaas 1l aculs .I dno-r8qns lerrrrou Surpuodsa-r:oc aql Jo arnlcnJls aql aururJalap o1 ,puocas
papaou aq plnol\ luarnnS-re uorlcnpur
pue lasec etur.rd ^{1aa11e1ar aql roJ
l"ql ol sno5eleue C roJ waroaql uorl€z -rrolce; Jo pul{ auros qsrlqelse ol 'lsrr} :s1rrd o^tl olur sapl^.lp rualqo.rd aql l€ql seas ,{14crnb euo 'g puE V Jo srapro aql uo uolldrunsse ou qlld\ 1nq 'cr1a,lc g pu€ V qll,t\ 'C sdnorS-ygy Jo ornlcnrls aq1 Bur>1ce11e u1 'dno.r8 uoru.ralenb aq1 o1 crqdlouosr d
pue
'C t (dg)N 'Z = d asec
;o asodsrp
o1 pa.rrnbor
sr luaum3.re pa,rloa .C -ur raqlEr V 'uorlcnpur ,tq snolloy osle ;oo.rd aql = (dg)N JI .uollcnpur aq1
ruroJ aql yo y dno.r8qns .}Iasl, oluo dg sderu 6 olul c I"rurou e sr detu raJsu€rJ ar{l Jo laura{ aqJ ,C (dg)t{ pu€ ppo s.l d Jo raJsuerl or{l pue uorlcnpul dq uerlaqe sl d ,{q speaco.rd ;oo.rd aq1 .rou pu€
'g>Ig qllltl VISV
=
'g;o
dno.r8qns,noldg-d e st (dg)N yo
6 dno.r3qns moldg-d
:(dg)N Jo roluac
lv jo dv dno.r8qns
owos roJ oo"oo
JI
e
dg 'z
.r,
"q1 ""11 urroJ ar{l ;o dno.r8 e sl (dg)tt
lBql
dg dno-r8qns no1.{g-d e raplsuoc a^\ 'V €I U (V)N yo
<
't .I
^\oqs
pu€
(V)N ,raqm
'rapJo ppo Jo rusrqd"rowolne ue Furllrru -p€ put Z rapro;o dnotBqns anbrun e Surleq dno-rE-g uerlaq"-uou dluo
aq1
aql 'Z = dy1 apqn lcrladc sl d leql sagldrul slql ,ppo sl d II 'd ,rap.ro 1o dno"r8qns anbrun sBr.J 1l uaql ,aldurs d11ecr1sr.ra1ce
s1 dno.r8 uor"ralenb
"
-r"qc lou sl J Jo 6 dno.r8qns mo1,{g-d B JI ler{l s.roqs luarnn8"re lecrlaun{lr -rB uV 'paqsllqelsa dlrsra sl J Jo ,{cuplodlru aq1 'sluaruale paxl} I"rArrl -uou lnoqJr^l sl p y1 'uorlcnpur dq snolloy J Jo arnlcnrls aq1 puz ,;, ;o raluac aql ur adE asaql 'sluauala paxrJ IErArrl-uou seq g aql Jo snql sl
J
y1
.(fq)
Jo luaurala ,t.rena 'e dq uorle8nluoc dq pacnpul
,f
ru.roJ
J Io ruslqd
4 n 'r_ B[qrB urroJ aql Jo C Jo sluetuala oql ar€ I JosluotualaaqJ'Cu! I"wrouJpuEI = I U Vqll^\J.v=Cacuoqpu" -rouroln? aql solouap
the second of group of lower order than G, the author decided to study these Problems
first.
If we assume then that
an ABA-group G contains a normal subgroup
TsuchthatG=ATandAnT=l,andifweletdbetheautomorphism easy to show of T induced by conjugation by a generator of A' it is very thattheremustbeafixedelementginTandafixedintegerrsuchthat (g) " ' ,"(j-l)t(g)) every element of r is of the form 'trt (*) dt (g) d'Zt for some i
and
j. If r = 0, T reduces
to the form studied in the relatively
prime case.
anci
a d -group' We have called a group T admitting such a representation only the identity have called the integer r the index of T. If / leaves
element of T fixed, we say that T is a groups is easily reduced to the study
regular {-group' The study of 4'of regular l'-groups' The main re-
sultforregular{-groupsisthattheyareeitherabelianor2-stepnilpotent.Theproblemisverysimilartothatofgroupswhichadmitanauassumption is made tomorphism of prime order. However, in our case no dealing with a very speon the order of d; but on the other hand we are
cial class oi grouPs'
lf
vr is the least integer such that
g a r (s) ' ' ' 6(w-1)r (g)=1'
we
forms the clearly have O( d )w >' O(G). This inequality on the order of G
L:;isisofmanyofcurarguments.ourproolisdividedintotwoparts: lir.stshowingthataregular.l,-groupisnilpotent'andthenshowingthat
aregular{-groupofprimepowerorderiseitherabelianorofclass2' unknown to us' we were Slnce Thompson's resrllts on p-normalitylz were as well as the soto treat the general case of regular
Iiirced
'6-groups general case is no longer called non-exceptional case (the argument in the necessary) '
S.rrthenon-exceptionalcasethe''leastcriminal''isasusualagroup (p'p'''''p)' Q is abelian of the form T = PQ, where P is abelian of type 1B
6I
'JV aql sI
J
pue
'J *gV = f
'J
*g ,I = J U (V) *N aJar{.t\
qlr.{\ asr./r\luawalo salnruuroc
(V) *N = C leql qcns
;
Jo dno.r3qns rolelnrutuoc
dno.r8qns
l"tnrou allc,{c € suleluoe
C uaqJ '(V)rtt Jo punoq.raddn eq1 = *BV = (V) *tt las pue ((V) (V)rN uollcnpul ,{q aurJac 'c11c,tc are
g
pue
v
araq/n
'gy
,_rp)n
=
= C leT
g:paulBl
-qo
d11uaca"r
a^"q a.r qclq^\ uraroaql
aql ul papnlcur ar€ sllnsar
8ur.,rro11oJ
asaq;,
'S'ntr'V aq1 ur .reedde III^\ V = (V)N as"c aql w sllns ,'sFurpaeeord -ar aqJ 'c11c,tc g pue V qlln sdno-r3 gV Jo asec lelcads aq1 ur uollsanb Surpuodse.r.roc aql pa1e8r1saau1 ror{lne aql pue uralsroH ,sdno;8-ygy
roJ ruaJoaql uoll€zlrolcBJ
" Jo
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.rado.rd ou prre
Certainly the results on AB-groups and on ABA-groups with A, B cyclic of relatively prime order suggest the factorization theorem which
for arbitrary ABA-groups in which A,B are cyclic. However, there exist ABA-groups (all sotvable) which do not one would hope to prove holds
admit such a factorization. This class of "exceptional" ABA-groups appears to be very small, but its existence complicates the problem. At the present time a large number of partial results have been obtained,
but many details remain. As in the relatively prime case, the transfer is an essential tool. Theorem B of Higman-gattl0 also plays an important role in these results. Although the theorems on ABA-groups
rely in general
on
their spe-
cial properties, many of the results needed are of a general nature. It is hoped that the results and techniques involved in these problems
gest a method
for attacking a conjecture which Herstein has
group which contains an abelian
will
sug-
made that a
or even a cyclic subgroup which is its
own normalizer cannot be simPle.
BibliograPhY
J. On Finite Group-s-wit\?-q4gpqlaunt Generators I, F.oE. Nit. ecad. Sci. USA 3? 604-610 (1951). On Finite Groups with 2 Inde.pendent Generators IV, Proc' 2. NaL-Tcad. Sci. USA 37 808-813 (1951). 3. Feit. W. On the Structure of Frobenius Groups, Canadian Journal of Math. VoI. 9, 58?-596 (195?). 4. Gorenstein, D. A Class of Frobenius Groups, canadian Journal of Math. Vol.' 11. 39-4? (1959). Finite Groups which Admit an Automorphism with Few Or5. in Canadian Journal of Math. appear Eits;lo 1.
Douglas,
6.
and I. N. Herstein. A Class of Solvable Groups, to appear in eanTd-ian Journal of Math.
7.
andl.N.Herstein.ontheStructureofCertainFactorizable
Croups I, to appear in Proc. of Amer. Math' Soc'
20
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'8
ON GROUPS WHICH CONTAIN FROBENIUS GROUPS AS SUBGROUPS
_t
by Walter Feit'
Let G be a finite group and suppose that G contains a proper subgroup M with the following properties.
(i)
No element in M distinct from the identitv element commutes
with any element not in M.
(ii)
For everv element x in G. either xMx-1 = M or xMx-1 0 M
={+
If
a subgroup M of G which satisfies conditions (i) and
(ii) is normal in
G, thenby definition G is a Frobenius group' and M is called the Frobenius kernel of G or the regular subgroup of G. Let N(M) denote the
normalizer of G. If N(M) = M, then a classical result of Frobenius states that G is a Frobenius group and the Frobenius kernel K of G has the property that G/K is isomorphic to M. In order to avoid the case that
G
is a Frobenius group, we will also assume that
(iii)
rvr*l_
NIMLl_s.
For groups G containing a subgroup M that satisfies conditions (i), (ii), (iii), we will state a result which under certain conditions yields a relationship between the irreducible characters of N(M) and the irreducible characters of G. This result has several applications and the remainder of the paper is devoted to a discussion of these applications.
Let M be a subgroup that satisfies conditions (i), (ii), (iii). The following notation will be used. The order of M, N(M) wiII be denoted by rn, qm respectively. The assumptions imply that 22
q > 1 and q divides
(m-1)'
8U
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llnsar pouolluau a^oqe aql alels
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^lou
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lsBI
slqyf I
-raqrrr\jJ '(nr)u ro rarc'rtsr{c
=
t: ;r dluo pue JI *[t ue
I.7
,g < r .ro;
=
*l)
arorrl
01 reql "ro1io""j "1 ^roqs lInclJJIp lou sl lI ',{1aal1cadsa.r \ '*l: dq palouap aq III/n I7 ,{q pacnp -q (n)N 'g ;o .ra1ce.r€qc aq;, '1" = (f )l ? lal pu€ ,.ra1ce-reqc Ier^rrl aql s.l o; e"aq^'11 ;o s.ralcereqc alqrcnparrr ar.{l aq {: ' ' ' '1 ,o) IIB 1 laT
been done by M. Suzuki and G. E. Wall.3
It follows flom conditions (i), (ii), (iii), that N(M) is a Frobenius group with Frobenius kernel M. Hence the fundamental result recently proved by J. Thompson4 implies that M is nilpotent. Starting with this
fact, it is possible tomake some estimates of the values of the charac-
ters of M, and hence of the characters of G induced by the characters of M. The details are rather technical and will be published elsewhere.S The annoying hypothesis (v) is needed to show that M has sufficiently many characters of degree one, There seems to be no reason why the conclusion of Theorem 1 should not Le valid without hypothesis (v).
If
this hypothesis could be removed, it would considerably strengthen some of the applications of Theorem
1.
The methods used to deduce the next two results from Theorem
1
are similar to those used by R. Brauer, M. Suzuki and G. E.WaIl6 to de-
rive analogous results from the special case of Theorem 1 in which M is assumed to be abelian. Let G be a permutation group which satisiies the condition, (*) G is a doubly transitive permutation group on m+1 letters in which no non-trivial permutation leaves three letters f
ixed.
It is an easy consequence of condition
('l') that the order of G must
be qm(m+l), where q divides (m-1).
Theorem
F)
2.
and assume
Let G be a permutation group which satisfies condition
further that G contain@
(m+1).? Then m = pe
for some prime p, md iuo'?n,l . 4q2,where
is the Svlow p-group of G. Furthermore if S'o
= 11j
sn
, there exists an exactlyE triplv transitive permutation group Go with [Co : C] < Z. o
The latter part of this Theorem is known, more generally 24
it
has
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I{
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uaaq
cyclic. Hence Theorem 4 immediateiy yields
Corollary. Let
G be a nq4:qyclic simple group. Then G satisfies
the assumptions of Theorem 4
if
and only
if
G
is isomorphic to SL(2,24)
forsomea >1. Theorem
5. Let G be a non-cyclic simple group
of even order whose
Sylow 2-group is abelian. Assume that the centralizer of every element
of order two is nilpolg4!.,T-eq G is isomorphic to SL(2,24) for some
a >1..10 The assumption in Theorem 5 that the Sylow 2 -group of G
is abelian
is essential, since the simple group PSL(2, 7) of order 168 is not isomorphic to SL(2, 2a) for any a, yet the centralizer of every element of
order two in PSL(2, ?) is nilpotent. Another easy consequence of Theorem 4 is the following. Theorem
6. Let G be a simple
group of order 49', where g' is odd.
Assume that the intersection of the Svlow 2-group S, with any other Sylow 2-group of G has order one. Then G
is isomorphic to SL(2, 4),
the
simple group of order 60. There is some reason to hope that
it might
be possible to classify
all simple groups of order 49', with odd g', by these methods. However there are still some serious difficulties to be overcome before this can be accomplished.
Theorems 3,4,5,and 6 all depend on the fact that groups of even or-
der are in some ways easier to handle than groups of odd o"dur.11 We now state a result of a slightly different nature which does not use this fact. We
will consider groups which satisfy
the following condition.
(**) The groupG has order g = m1 . . . mk. k > 1. For each i = 1, . . ,, k, G contains a subgroup M, of order m, which satisfies conditions (i) and (ii)' 26
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faf
s1
C.reqfta
.Z tua"roaqf,
9. See the
first paper mentioned in footnote
3.
10.
In the second paper mentioned in footnote 3, M. Suzuki proves that itie g"oup" Sltzi Za; with a > 1 are the only simple groups in which
11.
For a systematic treatment of gro]ps of even.order, s-eg-R'. Brauer ind K. A. Fowler, "On Groups of nven Order," Ann. of Math., 62 (1955), pp. 565-583.
t2. Cf. M. Suzuki, "The Non-existence of a Certqin-Typg of SiTf le-. Groups of Odti Order," Proc. of the A.M.A., 8 (195?)' pp' 686-695.
28
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to Wielandt's notes [3] at Tubingen.
2.
Permutable Products. Traasitive Extensions of Permutation Groups' A and We say that a group G is the permutable product of subgroups
BifG=ABandAnB=l.CIearIyAB=BA.IfeitherAorBisnormal inG,thenGisthefamiliarsemi-directproductofthesegroups.Butin generalneithersubgroupneedbenormal.IfwerepresentGonthecosets may of B, then A is given its right regular representation and the Ietters of G, be regarded as the elements of A. If B contains no normal subgroup as a permuthen the representation of G is faithful and B is represented
tationontheelementsofAdifferentfromtheidentity.ConverselyifG isapermutationgroupwitharegularsubgroupA,thenBbeingthesubgroup fixing a letter, it follows that G is the permutable product of A and B.Wenotethatapermutationgroupoiprimedegreeisnecessarilyof this tyPe. In the study of groups with regular subgroups wielandt [z] iras intro-
ducedwhathecallsanS-ring(SafterSchur).LetAbetheregularsubgroupofG. Forafixeda eAletab,=b5(i)ux(i),i - 1, ' Nasbt ranges over the elements of B. since 1a = a1 it follorvs that a is in the = Sta) is determined by any element in it; set of aOlr;. The set
{tOtrl}
namely, S(a) = S(aO1r)), if ut
(i) '
S(a). In fact, S(a) consists of the letters
ofatransitiveconstituentofB,regardedasapermutationgrouponthe
elementsofA.InthegroupringA*ofAovertheintegers,letusput s(a)
= t, e g
g. Then the s(a) form a basis for a subring of A* and this is
S(a)
tne S-fing. U
e is Abelian,
(s (a) )P =
:
then
ae S(a)
it is easily
seen that
for
any prime p
mod P' s 1a,P) r
Using the S-ring, Wielandt has shown that
if G is primitive
and
if A is
AbelianofcompositeorderandhasatleastonecyclicSylowsubgroup'
thenGmustbedoublytransitive.IfAisofprimeorder,theneithera 30
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u.l H Jo luatuala
lnq paurrrrralap dlanbrun tou st Zq (l) ut
'H'Zq'q qtt,n Zqxlq = x{x,J I q,H ) q rod (t) 'usrqd.rouolne Jauur rrB sr arenbs asoql\
g
uo rusrqd.rourolne uB socnpur x pu€ .{
I_
(r)
ft)
'''' erall 'd r
-xJ
lua^a due u1 .z Japro
(r,o)
=*, {",*;'=t;
aq ol
x a{q uaae ,{eru
(r)
ain sdno.rS J0 zx elrur; roJ pu€ 'I pue 6 Sul8ueqeralur x lueurala rre u1€luoc 11llr a^ll.lsu"rl ,tlqnop 8ulaq g ueq,L .I rellal Bu1x1y 11 yo dno"rSqns aql aq d la"I
"
:de.tr SurtrolioJ eql ur ualqord uorsuelxe .srallal aql alElnr.uro; ,telu aA\ Surur?ura.r aql uo alrlltsu"rl puE rollal 3
Surxr;
11
dno.rEqns
alrlrsu€Jl
alrlrsu€rl
Jo uorsualxa
"
^{1qnop .d.ra,ra 'es.rnoa
;o ,pw
"
sE
papr"Sar eq deru p dno.r5
,aar1rsue.r1 dlqnop sr
g uaql
,g uo
a^lllsrrerl sl H II 'g dnolSqns € Io slasoc aql uo paluasa.rda.r dno.r8 due dew
II"c (l
p ueql 'H uo pesodrur sr uorlrpuoc
ou JI
.H
Jo uorsualxa
alrlrsu"rl
a^t uaql 's.ra11a1 Sururerua"r aql uo H
"
aq C
sl 0 Bugxr; dno.r5qns eql q?1q.4\ ul = ,s las aql uo dno.r3 uorlelnrn.rad E sl c JI pue s 01 0 raual raqlrnJ
10'si r ulofpe aA\ II
'S srallol Jo las € uo dno.r8 uorlelnru.rad € aq H laT
'g
sdno.rS
alrlrsu"rl dlqnop aq1;o
aEpallrourl
JaqlrnJ osl? pu" g luarualduoa alllrsrr"rlul u€ Jo arnlBu aql uo uorletu
-rolur arour ol sureurar
aABq 01
r1cnl^I
alq€rrsap aq pFo^\ 1I .slFsor asaql pualxa
01 auop aq
'allllsue.r1 dlqnop st C ro Ierurou sr dno.rEqns_d ,troldg
Finite groups of this type correspond to nearfields, and Zassenhaus [1] has given a complete determination of these groups. Zassenhaus [Z] iras also considered the problem of finding transitive extensions
of
ertain
collineation groups. He considers collineation groups H of projective or affine spaces over finite fields requiring that if the space is of dimension n that H should be transitive on sets of n independent points. He finds that
only one.such group possesses a transitive extension. This is the
little
projective (i.e. unimodular) collineation group of the plane with 21 points, and its transitive extensions lead to the Mathieu groups of degrees 23 and 24 which
will be discussed later.
Holyoke
If]
nas given the conditions on transitive extensions in detail
and has shown in particular that
if a simply transitive group H has at most
k extensions to doubly transitive groups which are permutation isomorphic then it has an extension with further letters to a group at most (k-2)-ply
transitive, naturally excluding alternating and symmetric groups.
3. Multipiy Transitive
GrouPs
The problem of the existence of groups which are more than
triply
transitive remains highly mysterious. Apart from alternating and symmetric groups, the only known quadruply transitive groups are the Mathieu groups on 11, 12,23, and 24 letters, those on 12 and 24 letters being quintuply transitive. There is an infinite family of triply transitive groups' the
family of semi-linear fractional groups over a finite field F:
**axl+b,ad_bc*o cxo+d
where
x +- xs are automorphisms of F. Certain subgroups of these
groups
are also triPly transitive. In discussing multiply transitive groups, we shall exclude the alternating and symmetric groups as
trivial for
these purposes.
It was loown
to Jordan that a group of degree n could not be mole than (n/s + 1)-ply 32
88
'alr11sue.r1 ,{lr.ressaceu 1ou q3noq1 'H Jo (d)S dno,r8qns-d no1.{g e ,a:oru.raq1
-rnd 'saop tn aculs H Jo rapro aql sapl^Ip d uaql'ru = l-u 3u1p1,rrp aur.rd e sl d JI 'sJallal 1-u Surureruor or{1 uo alrllsuerl pu" srallel 1 Surxr; g dno-ri -qns € s"q C uaql 'aar1rsue.r1.{1d-(1+1) sl C JI 'sllnser asaql Jo uorssncsrp "ro; aler.rdo.rdde a.re sdno"r8 anllrsuerl ,{1d-1 uo s{rBurar le"reua8 aruog
'a^rJ.suEE-TiA-l wq1 a"roru aq louuec u aaTEap yo p
'ZI< r'27'(1-.r;"r t rl'Zd >I'r+d{-uyy 'sas€a .,{ueru ,{lalrulJul
q
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moi5T
ueql
'(-ra4.re6.g) ruaroeqJ
l 01 relrurrs sr Ja{r€d lsauJg
luocal V .suorlcrrlsa.r,ra11aq sear8 rrreroaql s,JaIIrIt qclql\ JoJ u Jo sanl€A fueru dlelrurJur ar€ aJarn acurs
ol anp'peqsllqndrm
's,-re111141
opnlcur
1ad se'rna.roaql
'.raaa.4r,oq '1ou seop
llnsar s,lpueIariA .Zt00g.
= g Sof
f
+
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'Z
- yy't <
1 roJ
a^Illsu?rl ploJ-l oq louuEe u ae.r8ap ;o dno.r8 e - x > d > z/x lE^ralur aq1 u] d ernlrdB slslxa
zI < u ro] 13q1 s/r\oIIoJ ll'u a,raq1 '2, < x raqunu 1ea.r due roJ l€r{l 'a'1 lalelnlsod
s,pue.r1,rag Fursn
'dnorJ crJJauurAs Jo;url"uralle aql sI 1l ssalun a^rlrsuerJ pIoJ-(I + r) sE tlanul sE aq louu"c u a5.r5-aa yoffioTllE 'Z = t'T = 1.roy ldarxg '>l < r '>I
< d putarur.rd
€ st d oraq,t .r +
'lrrurl rallaq qcnrrr
d>1
=u
B sear8 9161 ur
laT'(.rantni'y'p) [1]
ra11r1n1
'V'C
rna"roaql
Jo uraroaql V
'dno.r8 cl.rlarnruds .ro 8u-r1eu"ra1le eql sl C acuaq^\
pur; d.ressacau Jr ssoaord
p
dno.r8 aq1
srellal t/u6 u"ql ssal Surceldsrp luawala uB r{111r\ 1.re1s ,fern a^\ I + g/u < I JI 'srallal ur alcdc oerql
Z+\Z "
-ar
uaql
E
sZ ueql arour ou Surceldslp prmoJ 's.ra11a1
pue dsea
aq1 3u11eade.r pue
aq,{eu;yr_1,I_V roltlnturuoa
s ,{11cexa saceldslp V luatuala u€ Jr leql Surrrroqs uo sarl
sr;oo;d eq;, 'ZI = u roJ
paur€118 Suraq 1rwr1 srql'ear1rsue.r1
displaces all the m letters which H displaces since a subgroup of H
fix-
ing a letter has index in H a multiple of m and so its order is not divis-
ible by the highest power of p dividing the order of H. The elements of
t letters fixed by H into themselves form a subgroup K which is intransitive and has S, the symmetric group on t letters as the constituent on the t letters fixed by H. Here H < K and [X:ff] = tt' G which take the
Theorem. Let G be a t-ply transitive group on n letters. Let H be a subgroup fixing t letteqs a4! !q! Q be a Sylow q-subgroup of H, where Q fixes
w > t letters. Thenlllg,nq4qaliZel-in
on tJre w
G oi Q
is t-plv transitive
letters fixed by Q.
Proof: Let
a1,
, &t and
b1, . , bt be two ordered
sets of
t let-
ters both sets being from the w letters fixed by Q. Then since G is t-ply transitive, there is an element x of G taking a, into br, for i = 1,..', t. -1 -1 bt and Q and x ^Qx are Sylow subgroups of Then x-'& fixes br, ,
the group fixing b1, . . . , bt and so conjugate in this group. Hence for some y fixing
br,
,bt
-1 -1 we have y-^(x-^Qx)y = Q. But then z = xy is
an element in normalizer of Q taking
?1,
, a, into b1, . . . , b, in this
order. Z must, of course, take the w letters fixed by Q into themselves. From this it follows that the normalizer of Q is t-ply transitive on the w
letters which it fixes, proving our theorem. Let F be an intransitive group on n letters and let n = r + s where one set of transitive constituents consists of r letters, the other of s letters. Then F may be regarded as the subdirect product of a group A on r letters and a group B on the remaining s letters. If At is the subgroup of A which occurs with the identity of B, and Bt the subgroup of B which
it is not difficult to show* that At 4 A, Br 4 B and that A/ Al = A/Bt. In particular if A and B have only the iden-
occurs with the identity of A, then
*Marshall HaIl, Jr. [2] p.
6S.
34
9t
-t t/ft->l)
{y
,re qcttg 'saceldslp
l! srallel 4 aq1 uo dno.rE ly yo dno.rEqns ? 'e'l '{y auo fllcexa Eu11eu.re1p aW prre srallal r1-1 3u1x1y suleluoc g dno.r8qns arn g,/(I + lf) < { paxtJ E roJ l"r{l reqlrnJ asoddns a6 'g dnorEqns ? Io slasoc ,ro ly ;o uorleluasa.rdor ar{l sl g uaql ily oi clqd (f
-q)
-roruosl sl u pu" I"rurou
"
eABq
su1?luoc
>r
I
= N ler{l 1s,r1; asoddnS
lsntu
g '1ca.rrp lou sl €J pue lv
'lg/tll=
a1dw1s sI 111aculg
'lV
= N/S leql qcns N dno.rtqns yo lcnpo.rd laarrpqns aql pu"
trlN < II t"qt serrnbar slr{l t"r{l Eullroqs u1 slslsuoc;oo.rd s,lpuelel1t'srallal W eql Jo erour.ro;1eq Fur
-xlI g
Jo sluawale
1.rer{cog aq1 ,{ys11es
Iu qtr^\
tll
=
qll/{ A\ ur pa.rled 1ou ly;o alcdc-g e 1eq1 'uo-rlrpuoc "1 ol repro ur 'asoddns r(srn an pue ,€I pu" lV ;o lcnpo.rd
laarrp aql lou sl slr{l
6 yo flrpurulru
eql dg 'g pue
ly
;o lanpo"d
1ce.r1p
srellel I + ru aql uo pu" srallal ru des uo g luanlrlsuoc a^Il -Isrr€Jl rer{lrn} s"q /r\ usqJ, 'I"rululru 16 esoddns deu ait\ srallal 1 lsrll
-qns aql sl
aql uo
-rell"
\
71l
"
a.re qatq/r\ C yo sdno.r8qns
aq1 aq
-a.r r{.reae Io
I
II" JO 'dno.rE cr.r1auu,{s
.ro Eurleu
lsnul C lreqcog Io uaroaql aql fq uoql luenlllsuoc Eururern
srallal aql
lsr-rJ aql uo
ly
Jo arortr
q"1q^
"1 a^-rlrsu"rl dtd-t aq C
ro l1eq
11 dno.r8qns
ieT 'lpuelalt\
"
sa*1y
ly
uo aladc-g B II
'srallsl
'srallal u uo , ? I aql 01 urnl sn laT
seq C uaqJ
Jo uroJoer{l
'srallaI ne; e dluo 3u1ce1dslp sluauala ureluoc ,{eru sdno.rS aal11rn1.rdrul 'aslnoc
Io 'puv 'a1cdc o^11 B pue elcdc ae.rq1 e dlal-rlcadsar u-reluoc qclqm sdnolE cr.rlewwfs pue Eulleuralp aql ol lserluoc 3u14rr1s ur sdno.r8 alrllsuBrl .{ldn.rpenb yo $.rado.rd E sllqlqxa lror1co€J Jo waroaql aql leql alou
ar16
'sratlal I - f?- tsBaIJr sa5"dfl p lueluale AJeAa Alrluept arn roJ lclacxa 'clnoJb errlauur^s Jo SurleuJall€ or{}
u
'(l.ra{cog) ruaroar{f
'[t]
traVcoA Io uaroaql arfl Io asn a{Bru osIB lpuelalA,\ put ra{J€d
1ou-
'1cnpo.rd
-qns rlaql pue E =
lcarlp rraql paapur
Ig 'V = IV u"ql aEerur clqd.roruoruoq
s1
lcnpotd lcorrp
uouuroc e se
,{1r1
cycles. From this it fotlows that a given 3-cycle is contained in
MFF+itB
conjugates of
I{. This is the number of letters fixed in
the
representation of the 3-cycle in the representation B of A1, being the representation of A, on cosets of H. For Least M/2 contrary to our choice of the
k
> (4t +1)/5 this number is at
transitive constituent B of W to
satisfy the Bochert condition. with a number of intermediate results, Wielandt shows that essentially this situation must hold for any constituent B on
M < N(t) Ietters
even when B
is homomorphic to A, and not
necessarily isomorphic.
Miller's result is more easily obtained. With n = k p + r, r > k, p > k, r > 3 a group G of degree n which is (r + l)-fold transitive contains a subgroup H fixing r letters and transitive on the remaining kp
letters. A Sylow p-subgroup P of H must displace all kp letters and as k < p there are less than p2 lutt""" and so P is the subdirect product of k cycles of length p. U, the normalizer of P is the symmetric group
first r letters. U has a subgroup V of index 2 which is A, on the first r-letters. Because P is simply a subdirect product of k < r pcycles it is not difficult to show that B, the part of V on the kp letters displaced by V contains no factor isomorphic to A, for r > 5. Hence V S" on the
is the direct product of A, and B. But as A, contains a 3-cycle, G contains a 3-cycle and so must be the alternating or symmetric group. For
r
= 3 and 4
similar considerations give the same result.
Parker's result is an elaboration of the Milter technique but depends ultimately on the Bochert theorem. Suppose G is of degree n = k p + r'
k < r(r-1)/2, r 212, and that G is (r + l)-ply transitive. Then G contains a subgroup H fixing r letters and its Sylow p-subgroup P has
k.
p2,
constituents of Iength p or pz displacing kp letters in all. Since a p-group of degree
p',
can be moderately complicated, the nature of P in this case
is far more complicated than in the case considered by Miller. G contains 36
,,8
'(4 s1 raqrunu aql tZy{ dno.r8 nagqle4 oql roJ) srailal
ro I
(g (p dlleexe sexl; 'g H;o dno.r8-6 noldg
'Jopro uaAa Jo f,11.ressecau sl srellal .rno;
e'Zd leql s^\oIIoJ Surxr; g dno.rSqns e g
II ua^a lI sdno.r8
ly olrlrsuerl fldn.rpenb (a]lqI) reqlo il" rog 'TI^ dno.r8 nalqleyq aq1 "o rlg ;o auo flr.ressacau s! rapro ppo alIuIJ Jo sl srallal .rno; 8ur '9V '9S -xgy dnorEqns 1eq1 8u1,noqs
"
qcrq/r\ u1 'e11u-tyu-t ,{1q1ssod 'dno.r8
llnser
"
e
[1] roqln" aq;, 'II^ dno.r8 na1q1ery dluo uea d1g1uap1 aql sl sralla1 .rno; Surxr; dno.r8
s-Iql papuaga seq
aql ro 9V '9S '?S -qns
e^rlrsu"rl dldn.rpenb
"q
qclq^t ul dno.r8 aa111sue.4 dldn.rpenb
a11u1y
B lern pa^\oqs
[6] uzp.rog
'eIqeIIB^" sI uolleruro;ul Jo lunoure pallwll e dpo '.rey os '1nq eq lsnru sdno.rt al1llstrerl dldn.rpenb uo suorlcrrlsar
(8uo.r1s
alllelllenb 'uorlBJal
-1e lnoqll^\ elqeclldde s1 droaql slr{ Jo
sI r{c1rltr
'6
Eurzrlerurou aJoJaq se
n
lsar ar{J 'srallal r lsrlJ
aq1 uo
dno.r8qns B u-rtluoc lsnur C
ry
&Irrlll
-srrBrl aldn.rpunb ;o sgsaqlodfq arn Japrm asrd\raqlo rod 'paleloll aq lsnrrr
?,/G- r) r > {'rU t nsuolllpuocarnr+d:I=uquaqlsrellalrsaxrJ pue H Io dno.r8qns-d ,troldg e sI dldn.rpenb aql Io
srallal
.rno;
d II pue u aa.rSep Io C dno"r8 anrlrsue"rl 3u1x1; g dno"r8qns Jo rapro aq1 saplllp d
Jl arag 'a^Illsuerl .{ldn.rpenb sl t{c!qr\
tZ <
"
u t{1l,vr (uS .ro uy 1ou; u aar6
-ap tso/trol ;o dno.r8 e .ro; Surqc.r€as ul uollcnpu-I ;o sasod.rnd roJ aq deru slql pu€
ppon
'VZ'tZ'Zl'lI f r teqt dluo ueau dern s1q1 1eq1 dldrug
a8pe1r\ou{ luasard .rng
,{ldn,rpenb B
l"rll
par.unssB
qcns roqunu €
'rg
sl srallal "o "y
sl r IBI{I asoddng
-penb o1 elqeclldde s1 d.roaql sIrI Jo lsow
.r uo dno.r8
allllsuerl
'sdno"r8 al11lsur"q [1dn.r
l"ql pa^rasqo sBrI ra{r€d
'rxra.roeql lraqcog aq1 o1 drz.rluoc
srellal
r lsrlJ eql uaql re{r"d '"v ol
u eql JI"q uer{l ssel 3u1ce1ds1p slueuele q€1uot }snw sJal}el
s1 dno.rSqns s,t,or{s = s 'sV sI qclq/$ A ;o " l"ql "o t5] crqd.roruosr dno.rEqns B u1€luoc lolru"c d Jo luanlnsuoc E Io srusrqd.roru -olne eq;, 'srellel r lsrll aql uo "y q"1q^ 6 Surzrleu.rou n dno.r8qns e "1
J. Tits If ] iras studied in considerable generality groups which are transitive on "independent" sets of k elements. His motivation is geometric and the properties of independence he postulates are those satisfied by k points in a k-dimensional space not lying in a proper subspace.
4.
Frobenius Groups
In 1901 Frobenius [1] proved that a transitive permutation group
G
of degree n, in which only the identity fixes as many as two letters, has a normal subgroup consisting of the identity and the
n-
1 elements
dis-
placing all letters. His original proof has been simplified, but aII known proofs depend on the theory of characters of matrix representations. The normal subgroup N may be regarded as given by its regular repre-
sentation, and the subgroup H fixing the identity of N as a group of automorphisms of N. The group H must be of a very special kind. A subgroup
of H of order
, p'or
pq is necessarily cyclic. Thus the Sylow subgroup is
cyclic or generalized quaternion. Zassenhaus It] tras determined all such groups explicitly.
is doubly transitive, then N is an Abelian group, and G may
When G
be regarded as the group of mappings
x +- xm
+ b, m
#
O
in a nearfield
K, N being the additive group of K and H the multiplicative group. The
distributive law (x + y) m = xm + ym holds but not the other. As we have already remarked, all finite nearfields have been determined by Zassenhaus. The author [2, p. fAZ] has shown that an infinite doubly transitive
Frobenius group is also the group of linear mappings in a nearfield provided that we also assume a Euclidean property; namely, that there is
at most one permutation displacing all letters and taking a given letter i into a given letter j. The structure of the group N in a Frobenius group has only recently been settled. able then N
It was
is
shown by W. Feit [f ] tnat
if we assume N to be solv-
indeed nilpotent. By developing some very deep techniques 38
6t uaql 'U
<
1t\
r{ll1r\ a^rlrsuErl d1d-lr s1 C
}I 'a^llrsur.r1
qc-rq.r ur sas?c aql 3ur1e311saaur uaaq s?q rornn?
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aqJ
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ul sl C (11) pue alrlrsutrl ,{1d-(1+ru-u) sl C a^rlrrul.rd st g lt (l) leql pa.roqs It] ueprol'saaeldslp H qJ-rq^\ srallal w aql uo a^rlrsu"rl pue 1uaaa,{ue
srallal tu-u Surxr; dno.r8qns
€ H pu€ u aa.rSap ;o dno.r5 B aq C lorl ',{1tado.rd srql
r{ll,n
,8
t
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.g.
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aa.r8ap ;o sdno.r5 rar.nrnJ pue 6Z9 ,lZT ,6V ,gT,,6 = u r{llrrr (I_u)u srapro ;o sdno.rS plallrEau aql apnlcul asoql .sdno.r8 a^Ilrsu"rl ,{1qnop aIqB^Ios
Jo raqurnu ollurJ ? dluo are a.raq1 'aseq1;o sdnor8qns pue sdno.rS asaql
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wo-ry 1.rede
'suolleluasa.rda.r yo f.ro
-aql
aql Jo ap?w sr asn a^lsuelxa ,11nsa.r
qclq,l\
0I
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sr{J 'I +
zru
"
slql rod .alrlrsuerl
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s1
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,{1qnop sr dg aa.r5ap yo dno.r8 earlrwl.rd B l"q1 udoqs
s"q [Z] lpu?Ieli6
sllnsag lelcadg atuos 'g
fetu suorlrpuoc raqlrnJ uaAa lnq ,g yo .rap.ro aq1 Surplarp arurrd d.raaa .roy (d)l ,iq paprmoq sl N Jo ssBlc aql 'lap.ro alrsodruoc ;o 'p1oq
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's=(t)r{'r=(z){ a.lag 'd uo dluo Eulpuadap .ra8alur ue 3u1aq slql ,(d){ lsoru lB ssela Jo sl N uaq; 'fi11uap1 aq1 dluo 8u1x1y d rapro Jo usrqd.roruoln€ uE sllwpe qclq^\ N dno.r8 luelodllu '1ua1od11u aq
"
Jo ss?Ic aql pe1e811saaur seq
II]
uern8lg ureq?rg
lsnw N ler{l Suuroqs ur papaacrns /tlou s"q
[t]
uosdruoqJ uqof
a subgroup G, of G fixing w-2 letters
will
be doubly but not
triply transi-
tive and will contain the subgroup H. Hence our original G is at most a
transitive extenBion of a group G of degree n which is doubly but not triply transitive and contains a subgroup H fixing n-m letters and transitive on the m
letters which it displaces. We now replace H, if necessary, with
a maximal group fixing
n-k
=m
k > 3 letters and transitive
on the remaining
letters. We now find that the sets of letters fixed by H and the
conjugates of H from the blocks of an incomplete block design D and that G may be regarded as an automorphism group of D. We 1+
r(k-l)
where
will
r is the number of times any given letter
have n
=
appears in dif -
ferent blocks. If b is the number of blocks (in G, b is the number of conjugates of H) we also have the relation bk = nr =
"2k
- ,2 *
".
Any pair
of distinct letters occurs together in precisely one block. Conversely, suppose we are given a block design D with b blocks of
k letters each, a total of n distinct letters, each letter occurring in r dis-
tinct blocks and any pair of distinct letters occurring together in exactly one block. The parameters must satisfy the relations
bk=nr r(k-1) = n
1.
will satisfy our requirements if transitive on the letters of D and if the subgroup of G fixing
Then a group G of automorphisms of D G is doubly
all the letters of a block of D is transitive on the remaining letters.
If k = 3 the design D is a Steiner triple system such a group of collineations only
geometry over GF(2) or
and D
will
possess
if D is the set of lines in a projective
if D is the set of lines in an affine geometry
over GF(3).
California Institute of Technology* Pasadena, California -_---T_This paper was written at the Ohio State University. 40
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REVIEW OF SOME RESULTS IN COLLINEATION GROUPS by D. R. Hughesl
1.
Introduction
ln this paper we will attempt to review a part of the subject of collineation groups of finite projective planes. We largely disregard the more or less classical aspects of Desarguesian geometry, and we also skip the infinite planes. An example due to Marshall HaIl shows that
there are infinite projective planes with trivial collineation groups (and
it is also probably possible to construct
an infinite plane with any pre-
scribed countably infinite group as exactly its full collineation group). But in the finite case, every known plane has a rather "Iarge" collinea-
tion group. Whether this is an accident of our limited knowledge, or
a
theorem yet to be proved, is one of the most important problems in the study of finite planes. Section 2 is devoted to basic material, including nonexistence theo-
rems and configuration theorems. In Section 3 we list the finite nonDesarguesian planes known at the present time, and in Section 4 their
collineation groups, so far as they have been determined, are given. Sec-
tion 5 gives some special results. Statements of results
il
the paper are either followed by a reference
to a specific source, or else they can be found in Pickert's book;23 the two papersl6'12 by HaII are also very good references.
1. Supported in part by the United States No. AF 18(600)-1383. 42
Air Force under Contract
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tions with the same axis clearly form a group; the subset of elations in
this group is a normal subgroup. We call d a (V, K) central collineation,
if d is central, with center
V, axis K.
Let V and K be point and line, respectively, in a projective plane
..
P, are points of z such that neither is on K, neither is equal to V, but W'PZ is a line; if for all such choices of Pt and Pr, there is a (V, K) central collineation such that P1 d = P2, then we say that ' Suppose
Pt
and
is (V, K) transitive. It is not hard to show that there is at most one (V, K) central collineation sending P, onto P2r so (V, K) transitivity amounts to demanding that the group of (V, K) central collineation be as large as pos-
sible: (simpty) transitive and regular on the "non-fixed" points of a "non-
fixed" line through V. Without saying what Desargues' Theorem is,
we
remark that o is Desarguesian if and only if it is (V, K) transitive for
all
choices of V and K. One of the most interesting and
fruitful studies
of
non-Desarguesian geometry has resulted from considering projective planes with only a limited amount of (V' K) transitivity.
For any projective plane, we can construct an algebraic system called a "planar ternary ring"; this is an analogue of the use of a field
to co-ordinatize Desarguesian geometries. We will not need to be concerned with the most general of these planar ternary rings, but
will
con-
tent ourselves with certain special ones which are the most important in
terms of present knowledge.
Let (R, +, .), or merely R, be a system with at least two distinct elements 0, 1 and with two binary operations, addition (+), and multiplication (written a . b or ab),'all satisfYing:
(i) (ii)
under addition, R is a group with "identity" 0; under multiplication, the non-zero elements of R form a loop
with identity
(iii)
1;
x(y + z) = xy + xz, all x, Y, 44
z
e R;
s?
cnm l<-
+ (o) (x) -+ (x)
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Then
{ is a collineation of r,
?.rd indeed
is an elation with axis
L-,
center (m) where ma +b = 0, if a + 0, or center (co) if a = 0. In fact, r is (P, L-) transitive for every point P on L-. This last property is characteristic of (left) V-W planes:
If o is (V,K) transitive for two points V on K, then z is a (left) V-W plane with K as L-, and thus o is (V' K) transitive Theorem 2.1.
for all points V on K. Also we have: Theorem 2.2.
If
is a (left) V-W plane with K as
ordinate rings with K as
L-
L-,
then
all co-
are (left) V-W systems, but not necessarily
isomorphic.
Let us restrict attention to the finite case' R is a proper (left or right) V-W system if it is not a field, and r is a proper V-W plane if it
is co-ordinatized by a proper v-w system. If one of the v-w systems of a v-w plane is a field, then all the co-ordinate rings for the plane are fields and the plane is Desarguesian. We have: Theorem 2.3. If, for two different choices of the Iine Lco (the point (co)
), the plane
r is co-ordinatized
r is Desarguesian.
So a
by a left (right) V-W system, then
proper left (right) V-W plane has the property
that none of its collineations move the line
L-
(the point (co) ).
finite non-associative division ring plane, then no collineation the line
ff r is a of o moves
L- or the Point (co).
This theorem reduces the study of collineation groups of non-associative division ring planes to a simply stated problem. Let R be a division ring, and a, b non-zero elements of R, T an additive one-to-one mapping of R upon R. Then the triple (T, a' b) is an autotopism of R
if
(xa)T(bv)T=(xY) T, for all x, j e R. The autotopisms of R form a group under the operation
LN
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II.
The Hughes planus.13 These cannot be co-ordinatized by V-W
systems, and are the only known finite planes with this property. Orders p2t, for aII odd primes p, positive integers t. Probably self-dual, but this
is only established 1o. o2t = 9; uniqueness still open. rIL The Hall V-W pl"n"r.10 Let F be the Galois field GF(q), q,+
2,
and let R be a two dimensional vector space over F, with basis elements
1, tr. Let f.(z) = z2 -
az
-
b be an irreducible quadratic over F, and then
define multiplication in R by:
(i) x(u+ Iv) =xu+ tr'xv _1 (ii) if y1O,(x+ \y)(u+ rv)=)m- y ^vf(x) + I(yu-xv+av). Roughly, this corresponde to demanding that every element of R, except-
ing elements in F, shall satisfy f(z) = O. The system R is always aproper
left V-W system and the multiplication is associative only in the case that F = GF(3) andl(z) = 12 *
1.
The right distributive law never holds'
Al-
though there are several Hall V-W systems of any fixed order q2, th"y
all co-ordinatize the same plane, and the plane is never self-dual; dual plane
the
is a right V-W plane co-ordinatized by the system anti-iso-
morphic to R.
W.
The Andr6 V-W planes.4 Lut R be a Galois field and
.l
a non-
identity automorphism of R; Iet K be the subfield of elements fixed by o.
xot1,*h"r"ti" ForeachXe R,define ,(x) =x xo'*o2'..' the order of a; then also R has degree t over K. Furthermore r (x) e K, aII x e R. Let p be any mapping of K into the integers modulo t, with only the proviso that
p
(1) = 0
= p (0). We define a system R,
addition is the same as R, but with multiplication given by
xoY=1Y wnefe dx=
x
c' su(x) 48
whose
6t
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Then R
a non-associative division ring.
is
VIII.
Dickson's commutative division ring planes.l When c = -1r the
division rings of type VI are commutative. But another class is known.
Let F be a Galois field with
r
a non-identity automorphism, and let R be
a two-dimensional vector space over F with basis elements
1' tr ' Multi-
plication is:
(x+ ry) (u+ rv)=(xu+ Eyo v")+ r(xv+yu) where 3 is an element of F. If F has characteristic not two, then R is a division ring if and only if E is a non-square in F; R is always non-asso-
ciative(if
IX.
S
+0).
We add here a remark that other division rings exist, whose na-
ture is not completely understood. Knowledge of them is fragmentary,
it
and
seems best to disregard them here.
4.
Known collineation groups We
wili list here, with references, some of what is known about the
collineation groups of the planes of Section 3. The Desarguesian planes (Class I) are of course classical, and we only remark that they all have non- solvable collineation groups.
II.
The Hughes planes have been analyzed by Zappa and Rosati.24'27
A typical Hughes plane n has order p2n, p an odd prime, and contains a
ro of order pn. Th" projective group of ro P€rto all of r I the fuII group of r is the product of this
Desarguesian sub-plane
mits an extension
projective group with the automorphism group of a near-field of order t^ p'tt. Thus the group is non-solvable. It has the additional interesting prop-
r into iust two transitive one of which is the set of points of ro.
erty of breaking the points (and lines) of stituents,
III.
con-
The HaIl V-W planes have also been completely analyzed.l? Th"
description of the group is somewhat complicated, but it is always non50
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5.
Other theorems about collineation groups Besides the study of the groups of specific planes, there consists a
large body of more general theorems about the implications of the existence of collineations. Two of the more basic of these are:
Theorem 5.1.
fi
,l' is a collineation of a finite projective plane, then
the number of points fixed
by ,f equals the number of Iines fixed by {
'
Theorem 5.2.8'15'22 If G is a collineation group of a finite projec-
. , N1 (N2) the number of transitive constituents of points (lines) of r with respect to G, then Nf = NZ. Many of the theorems concern circumstances under which r must
tive plane
be Desarguesian. Gleason and Andr6 have proved:
If for every pair (P, L) of the finite projective plane o , where P is a point on the line L (not on the Iine L), there is Theorem 5.3.5'9
non-identity (P, L) translation (homology), then
a
r is Desarguesian.
Planes with transitive collineation groups have been studied
for
some
time. singer discovered that a finite Desarguesian plane of order n has a
cyclic collineation group of o"der n2 + n + 1 which is transitive on points and lines.25 Subsequent *o"k11 has indicated that probably only Desar-
guesian planes have this property. (It is interesting that
for infinite planes
the correct conjecture appears to be that all planes with this property are non-Desarguesian!) Indeed,
if r
has a transitive collineation group, then
o is probably Desarguesian. Failing to prove this theorem, wagner have shown the following, which is
still
a high-point in the field:
Theorem 5,4.20,21 V the finite projective plane group G which is doubly transiiive on points, then G contains all the elations
of r
Ostrom and
z
has a collineation
. is Desarguesian,
and
.
Also, the following is a step in the direction of analyzing transitive planes: 52
t9
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point P and one line L, where P is on L. The above result has some interest in
its bearing
on
order 10, since
that is the smallest order for which the existence of the plane is in doubt.
Bibliography
1. A. A. Albert,
On non-associative division algebras, Trans. Amer.
Soc., vol. 72 (1952),296-309.
2.
, Finite noncommutative division algebras, Proc. Amer.
3.
, On the collineation groups associated with twisted fields,
Math. Soc., vol. 9 (1958), 928-932.
Bull. CalEutfa-[Iath. Soc. (to appear).
4. Johannes Andr6, Uber nicht-Desarguessche Ebenen mit transitiver Translationsgruppe, Math. 2., vol. 60 (1954), 156-186.
5. , Uber Perspektivitd.ten in endlichen projektiven Ebenen, Arch. MdTh., voi 6 (1954), 29-32. 6.
(1955), 15F16-0-.
, Projektive Ebenen iiber Fastkijrpern, Math.
2., vol.
62
?. R. H. Bruck and H. J. Ryser, The nonexistence of certainfinite projective planes, Canadian J. Mith.,'vol 1 (1949), 88-93. 8. H. P. Dembowski, Verallgemeinerungen von Transitivititsklassen endlicher projektiver Ebenen, Math.2., vol. 69 (1958) , 58-89. 9. A. M. Gleason, Finite Fano
planes, Amer. J. Math.,
vol
78 (1956),
797- 807.
10. Marshall Hall, Jr., Projective planes, Trans. Amer. Math. Soc.t vol. 54 (1943), 229-277.
11.
, Cyclic projective planes, Duke Math. J., vol. 14 (1947),
t2.
, Projective planes and related topics, CalU. Inst. Tech.,
1079- 1090. 19
54.
13. D. R. Hughes, A class of non-Desarguesian projective planes, Canadian J. Math., vol. 9 (1957), 3?8-388.
14.
, Regular collineation groups, Proc. Amer. Math. Soc.,
vol. 8 ( 195?)--T59- 164.
15. . Collineations and generalized incidence matrices, Trans. Amer. Math---Soa., vol 86 (1957), 284-296. 16.
. Generalized incidence matrices over group algebras,
IIIinois J:-NIt[h-.', vol. 1 (195?),545-551. 54
9S
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SOME FINITE GROUPS WITH GEOMETRICAL PROPERTIES by W. Magnus
1. Introduction This is an expository paper. The term geometrical refers to the classical meaning of the word, namely to those properties of a group of transformations which permit the definition of congruence in the corresponding geometry.
first attempt at characterizing a group of motions in terms of properties of the group itself seems to be a paper by M. DEHN1, in which The
he studies the abstract properties of the group of motions (including re-
flections) of the hyperbolic Non-Euclidean plane and describes an abstract characterization of other continuous groups. Although we shall not discuss projective groups,
it
seems appropri-
ate to quote here a paper by N. S. MENDELSOHN,2 the methods of which
are closely related to ours.* Mendelsohn succeeds in characterizing completely the plane projective groups by two sets of conjugate subgroups,
their normalizers and their intersection. (EIis methods can be generalized for higher dimensions.) The proofs of the theorems stated here are to appear soon in the extended version of the Ph.D. thesis of G. Bachman.3 We shall present his results with some modifications.
2.
Axioms
Definition 1. A group M is called a group of n-dimensional motions
if it satisfies the following
(i)
conditions:
There exists a set of proper subgroups Pt of M which con-
*I am indebted to Professor G. de B. Robinson for calling my attention to this paper at the Symposium on Finite Groups. 56
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observations lead to the following
Definition conjugate
2. Let M be a group which contains a complete set of subgroups Pi (i = 1, . m) satisfying the postulation of
Definition 1. Then the P, shall be called the points of a geometry f . The group M acts as a group of transformations on I' according to the rule that, for any element
t of M, the map t(Pt) of Pt is
defined by
t(Pi) = tPit-1. The d-dimensional zubgroups of M are called the d-dimensional subspaces
of t. ; in particular, we shall call the one-dimensional
subspaces
'lines'
and the two-dimensional subspaces 'planes'' A
point P, is called incident on a subspace L6
if
LO
is a subgroup
of
Pr. The number n is called the dimension of the geometry' The action of t on LO is defined.bY t(Ld) = Two sets
if
tldt-1.
:1 and t,
of points or subspaces are called p-Sqglgg$
there exists an element t of M such that
t(>1)=:2' Now elements
t of M are called the motions of the geometry'
From the axioms (i) to (iv) for M, we can derive the following Statement of properties
of t
:
(i) AnY two Points are congruent. (ii) Any two d-dimensional subspaces are congruent' (iii) Let P be a point and let Lo and ti be two d-dimensional zubspaces incident on P where d > 1. Then there exists a motionwhichleavesPfixedandmapsLoontoLj.Thecorresponding fact is true for two (d + k)-dimensional subspaces incident on a d-dimensional subsPace' 58
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tively and without fix points on the points of the corresponding geometrY.
obviously, Frobenius' Theorem cannot be extended to infinite groups since the motions of the Non-Euclidean (hyperbolic) plane provide a coun-
ter example. In the two-dtmensional case, a finite group M, of motions determines a geometry uniquely. We shall state this as
Bachman's Theorem:
If a finite
group M contains a complete set of
conjugate subgroups Pt which are their own normalizers in M and anv two of which intersect in the identitv, then the P, are unjggelX
determined by M. The proof follows from the result stated in Bachman,3 Theorem 3.2, and an application of J. G. Thompson's Theorem. Bachman showed:
It
M
finite group containing two subgroups P and Q, each of which is its own normalizer in M such that both P and Q have the property that each
is
a
of them intersects with any one oI its conjugates in the identity only, then
either P contains one of the conjugates of Q or Q contains one of the conjugates of P. Obviously, we may assume that P contains Q. We wish to show that P = Q. Let Nn anci Nn, resPectively, be the
normal divisors consisting of the identity together with the elements not in P or its conjugates and not in Q or its conjugates' If P contains Q, then Nn contains Nn. According to Burnside,4 the orders of P and Nn are coprime and the same is true for the orders of Q and Nn. Therefore' N^ must contain at least one sylow subgroup of P. According to Thompr.J
son, the group Nn
is nilpotent
and
its sylow subgroups are characteris-
tic in Nn. since Nn is normal in M, its sylow subgroups are also normal in M. Therefore, P must contain a normal divisor of M which is incompatible with the assumption that the coniugates of P intersect in the iden-
tity only. Therefore, P = Q. 60
I9
'l+3=lu o^€q a^\ tasec s1q1 uI 'paxrJ
sr J ) L
aJaq&
(/ -x)Zn = /. -.x suorlnlllsqns Jo sdnoJ3 aql arE IC
'(07")
pu" J ,'d 'o
".{1
aJal{/t\
d+*zo =-*
suorlnlrlsqns ;o dno.r8 aql aq ol uasoqc aq deru 6 dno.r8
aqJ 'tW dno.r3 e s1 3 .rapro Jo J plalJ slol€C
ur sluarclJJaoa qu^\
"
Z
aa.rSep yo (3'7)AT dno.r3 leuorlc€rJ r€aurl ,t.reaa aculs aldrurs aq deru try tlce; ur 18W uI suoll"Isu"rl;o dnol8qns € lslxa lou IIr^\ a.raq1 'ye.reua3 u1
'({*!)
l={oulq l"q1 pu€ 6 u1 ale8ntuot aJ?
lu'"'tI=!'Id
U
d=Iq
sdno.rE aq1
l"ql
pu"
6 slenba tw ut d Jo (d)N razll"urrou aql l"ql qcns 116 = 4 ku'''''I Ia sale8nfuoa sll pue 6 dno.r8qns e a^Bq a,n ttry dno.r8 e u1 = l) suorlo6 l?uolsuaunq-g ;o sdno;g allul.{
't
'rapro arul.rd
sl ;, ssalun uotuuloc
yo sdno.r8
u€ql aroru taurl t111,n tleraua8 3 uo aq s.{urn1e 111n slutod on1 saurl o^\1 lnq u1 arreq c11cdc Jo urns
lcarlp
aq1
u1 1u1od auo
i{uy'as.raaul sl1 oluo V Jo luaurale ,{:aaa sdeu qc1qo. , tuslqdJotuoln" aql dq rapro ppo ;o y dnorS crlcdc-uou tuellaqe u" Jo uolsualxa aql IUoJJ saslre aldruexa ue lsaurl aql uo d1ea111sue.r1 lce lou lllfi, ZW tle.raua8 u1';, yo
1 dnor8qns
c11cdc
l"tulx"tu "
Jo
sluauala aq1 qSnorql sunr Y aJaq^r
'-lsdrru Jo auo
sI qclq&) 6
1u1od
t
-uolsuaunp-odu Jo Zliq dno.r3
dq paullap aq B
u"r
ul suorl"lsu€rl
s11
au11
Jo
I
Y
dY
pue (!d sdnor8qns
t
aq1
'u,tou1 sI suo!1otu I€
dno.r8qns aql acuo
Since
LF(2,5)
=
LF(2,4i,
we also have an example of a group MU which has two different sets of subgroups P, satisfying
all the postulates for a group of three-dimen-
sional motions. BaehmanS has found necessary and sufficient conditions
for
an
M,
to have a normal divisor T of translations in the case where the D, are
their own normalizers in P. Geometrically, t}tis means that, if a one-dimentional subspace (a line) is mapped onto itseU in such a manner that one point remains fixed, the mapping
is the identity.
Let M, be a finite group of three-dimensional rnotions and let P,Pir D = DZ and D, be defined as above. Assume that the normalizer Nn(D) of
D in P satisfies Np(D) = D. Then
M, has a normal divisor T which acts transitively
points on the Pi
(i)
if
and only
and without fixed
if
The normalizer N(D) of D in
M, is the direct product of D and
, of N(D). (il) f,et ? run through the elements of P outside
a subgroup
jugates. Assume that the smallest subgroup T of aLL
D and its con-
M, containing d
and
n 0n-1 h"" no element in common with P. Bachman has shown that T
is always a normal divisir of Mr. If T has
no element in common with P, then
wttlt =.v. We shall prove now that conditions (i) and
(ii) are necessary. (That they
are also sufficient has already been shown by Bachman who also exhibited examples of groups MU with a normal divisor 62
T.)
Thus, let us assume
89
'suoll"cllqnd ra^og' gg61 palug.rd
eplsurng '1t\
-aU 'uorllpg pu3 'rapro allulJ Io sdno.rF ;o d.roaql 'OL
'Z'q1epsdnorS
a11u1y
'696I '6tt-99? u!€lrac u1 d.rlatuoeg '88
'0t 'Io 'sa1.rag prg 'lU uorl?as 'aog 'dog 'su€r; 'dno.r8
'gt6l
1e.raua8 arn Jo
-gg
u"uqr€g '0 '8
-/,t
'"pEu"C uorl"aurlloc anlcafo.rd
dery
uorl"zlralc"reqc crlaroaql dno.r8 g
'It6I g '1g.rr1ssp1;, 'l€W 'alrlawoaC
't
uqoslapuatrl's'N 'z
't8 .rap uaEel
uqac'w 'I
-punr5 upp ul ua8unqcs.rog aranau aElura raqan sacuaJaJou
'sdno.r8 3u11eu.ra11e aql roJ pue tB "rap.ro Jo pIaI]
sloI"D e u1 (3'ur)91 sdno.r8 l€uorla"rJ r"aull aql roJ anrl s! slql
'tr1nc11red uI 'uIN addl yo sdno.r8 se pa1o.rd.ralul aq u"c sdno.rS a1du1s
requnu
J,frJ
a3.re1
e 'Z
< u.ro; t1eq1 {J?uIaJ
eql
qlll\
apnlcuoc deru
yo
a7y1
{sllelap ;o .{1a1.rea e rod '*I ;, lerll = g'u€tuqc"g aas 'I ruoJ} slrtolloJ 1!'d p I78W aculg = d u*J'a.rola.raq;'s1ulod
xlJ a^Eq1ou op
| * L Jo sluatuala aql aauls I
l_u0u aq1 -Il"rus 'a.ro1a.raql 'pu€ l_,t 0,'
= dU;'d1sno1,rqg
aql pu€ B f,q pale.rauaS*I dno.rS lsa sdno.r8 aql Jo IIB ululuoo lsnru
I
a\oN
'd.ressacau s1 (1) (a.royaraqJ '(q)N =
eq1 slenba
q Suruleluoc
ex 0 al"q a^\'(C)H ut C Jo xapul !6 ;o aql aauls 'g x d lcnpo.rd lcarlp aq1 "aq*.tu
suleluoc (q)g'e.royaraq;, 'paxlJ d sal€al q searaq^\ paxlJ !d due aaeal
IIIII\
d lo T f
sluawala aq1 aculs
(q)N ut I€urou
I = qud
1e*"ou sI I acurs replsuoc pu" slsrxa I1"q1
pu€
sr d '^Ir"aIO '(A)NUI = d
84
1ou
u1
SYMMETRICAL DEFINITIONS FOR THE BINARY POLYHEDRAL GROUPS by H. S. M. Coxeter
1. Introduction. The binary
polyhedral groups arise in at least
four ways:
(1) abstractly. as the groups determined by the relations
Rl =sm=1r=RST, where
l,
m, n are small enough to make the order finite;
(2) algebraically, as finite groups of quaternionsl (3) geometrically. as.finite groups of Clifford translations in spherical 3-space; and hence
(4) topologically, as the fundamental groups of certain 3-dimensional manifolds discovered by Seifert and Threlfall.
After considering each of these aspects in turn, we shall derive some new abstract definitions ("presentations") which are symmetrical
in the sense that they admit an automorphism which permutes the gen-
erators. Some of these presentations were communicated to me privately by Dr. R. G. de Buda and Dr. B. H. Neumann, to both of whom I would ex-
press my gratitude.
2.
Rotations and quaternions. The finite groups of rotations in
three dimensions are the polyhedral groups ( l, m, n)
2.t
R/ =sm=tx =RST=1,
where
k=mn+n! + lm- lmn >0 64
def ined by
'uzO :arlcdc
=
't)
(u 'u
s1 dno.r8,,c11c,tc d.reu1q,, aq1 ,.re1ncr1.red u1
.[69 .d,g]
ISU=uI=*S= UE ,{q pauryap
rapro Jo (u -.ra1enb
Jo' (u'* 'tr)
dnot? 1e.rpeq,{1odXr€ulq eq1 sp1ar,{
t'Z
'\/wn
\/uwV
tr
V
Z rapJo
'ru '7 ) dno.r8 le.rpaqdlod aq1 :1ea.r8 s€ ocr^\l sr rapro asoq& suolu .suoru.ralenb 1o dno.r8 e sp1a1d suorlslor yo dno.r3 elrulJ r{cta snrl;,
lrrm pue suollelor uaal\laq acuapuodsa.rroc o1y\1-auo B sr araql ,uo11eru -roJsrr€rl aq1 Eu1ra11e lnoqll/rr e- dq paceldor aq uec e aculg '[Ogf .d 't]
gf
ursl{8a + gzd+
rlq + + f
soc = z
z'z
oJaqlr 'gxe olut x sruJoJsu€Jl 8d 'Zd sau.rsoc uollceJlp
qll/{ aull aql lnoq" / q3noql uorlelor
'Id
ar{l leql putJ aA\
'>1tx+tZx*rI*=" uolu.ralenb e"rnd eq1 dq
(6x 'Zx
'Ix)
saleulprooc u"!salr€C qlll\ lurod
aq1 luasa.rda.r am
y1
'1enba
lsnw olt\l Jal{lo o{1 '.,, s1 ,,413ue111,, lucl.raqds e yo a13ue auo yr '12q1 lD€J aql qll^\ pal"lcossB sl I = 7 uaq/tl u = ru uoll3lr1sa.r pa11drul aq;
aq
.>t/ulntr €InurroJ
'l '?,'t'V 'fu rapro Jo
qcea ur JapJo aq1
'0g 'v?, 'zl 'uZ 'u
Jo rapro Jo rapro Jo rapro Jo rapro
'u7 ate
{
=
(g 'g
dq ual13 sg asrc
}o sanl€A aallcedsa.r aql aculs
'z) (l 'g'z) = 'uU (g ', 'z) = 'uO = 1u'r'r', 'tg (u 'u 'I) = '?S
or{1
z
dno.r8 lz.rpaqesocr aql dno.r8 le.rpaqelco aql dno.r8 1e.rpaq"rlal aql dno.r8 le.rpaqlp aql dno.r8 cl1cfc aq1
:dlaweu '[gs 'gt-t
t
'66
,e1
group (2, 2, n) is dicyclic [4, p. Z]. fhe dicyclic groups include, as special cases, the quaternion group (2, Z, Z)
and the "binary dihedral"
(Hamilton's t2 = i2 = k2 = ijk) and the "generalized quaternion group"
<2,2, zq> [ro, p. rzo]. It is true,
2.4
though
tricky to prove [2, pp. 369-?0], that the relations
Rl =Sm=T"=RST=z
fmpty 22 = 1 (provided,
2, m,n, k are all positive; but if any of .{,, m,
n are negative the period of Z is greater than 2, and
if k = 0, Z is free
[2, p. sza]).
3. Permutations. The fact that the relations 2,4 do not imply Z = 1 is obvious for (1, n, n) , and well known for (2, 2, n) .In the more complicated cases
it
may be deduced either from the representation by
quaternions [2, pp. 3?0-?1] or from the following permutations.
for (2, S, S)
(degree
8):
R, =
(ac'a'c)(bd'b'd),
S= (bdab'd'a')(cc'), a = (abca'b'c')(dd').
ror (2, 3, 4)
(degree 16):
3 = (ae'a'e)(bc'b' g = (cahc'a'
c)(dhd'h' )(fg'f' g),
h' )(dged' g' e' )(bb'
)(ff '),
1' = (abcda'b' c'd' )(efghe'f' g'h' ).
For (2, S, s) 11
(degree 24):
= (aga' g' ) (bd'b'
d) (cf ' c' f
)(ek' e'k) (hj' h' j) (i .(,, i, .!,
1,
5 = (kdak'd'a')(cfbc'f'b')(jgej'g'e')(i Xhi',L'h'),
1
= (abcdea' b' c' d' e' ) (ghijkg' h' i' j'k' ) (ff' ) ( .t, 2' ).
In each case Z =
(aa')(bb') ., and the central quotient group is
derived by identifying the primed letters with the corresponding un-
primed letters. When this has been done, we have the ordinary tetrahedral, octahedral and icosahedral groups, representecl by permutations of the faces of the tetrahedron, octahedron and dodecahedron, which we
t9 -Enb
('e'r) suoruJal?nb llun
Jo 1s!suoJ
lsnur suoluJalenb Jo dno.r8
a11ur;
,{ue 's-ro1ee; aql Jo sruJou eql Jo lcnpo.rd aq1 o1 lenba sr suoruJal?nb o.rrl
;o lcnpo.rd aql Jo rtrrou aql acurs
'
1 e.rnSrg
q V o i'gg aal8ap Jo uorlsluosa"rda.r ou seq {l 'e '21 dno.rE T?;rpaqslco d.reulq aq.t! 'I arn8tg ur srue.rSerp 1a8e1qcg riq a1a.qsn111
ternions of norm 1). Since any unit quaternion may be expressed in the
*
form 2.2 (witn pf
* ni = t), any finite group of quaternions must be either polyhedral or binary polyhedral. The former possibility will arise
if
p!,,
-1 does not belong to the group of quaternions, in which case
the corresponding binary group is the direct product of the group of
quaternions and the group of order 2 generated by -1.
if possible, that ( l, m, n) is the direct product of (!,,m, n)andthegroupof order2generatedbyZ. Then (l ,^, n)must Suppose,
contain elements Ro =
RZr,
To = Tzt
So = SZs,
satisfying
*j=ti=4=Rosoro=1. This requires
| + rl
= 1 +sm = 1
which can happen only
if
+tn = I
.!., m, n (and
+
r +s +t
(mod 2), =0
r, s, t) are all
odd. Hence the
only case of a direct product is
(t,n,n) and the only
=
Gzn
3
tsn
x
G,
(n odd),
finite group of quaternions that is not a binarv polyhedral
group is the cyclic group generated by a single quaternion of odd period [0, p. ses].
It is, perhaps, worth while to point
out that
all the groups discussed
here are finite subgroups of the classical group A1, which is the continuous group of all unit quaternions [p, p. eZ6].
5.
The representation by points on a hypersphere. Representing
the quaternion
*=*o+xri+x2j+xrk 68
69
z'11 uorlaunrl at"tpotrrralut aql p1ar,{ s, {t'u't} { s 'l 'e } leco.rdraa.r on1 aIil pue'lvzt'aTt {"2} " {rt} lcnpo.rd .re1nsue1ca.r .ro usr.rd Ieuors - uarutp- t aq1 plard ur o.rt aql :[lZe 'd ,i] ru.rogrm r{po 1nq .reln5ar ", { }
lou sr lr q3noq1 'ado1d1od B prmoq aq1 '(a1dtu1s oo1
't
{e 'e '?}
sl
qctq/r'(r'r'I)
alnc.raudr{ aql prmoq
''3'a) adoldlod
1111s s1u1od
aql 1€ saueld.rad,{q lua8ur}
8u1pn1axa) sasec Eulureua.r aq1 u1
'f+ = Ix
I+ -lx'"
.re1n8a.r lecordrca.r aql prmoq a.raqds-g aql 01 saueldladfq
1ueSuq Surpuodsa.r.roc aql g aql ar€ daql qcrqn loJ
'{{ f ' ( Z,
'S ,Z
'g}
,?,)
,,adoilIod sso.rc,, aql }o sarlpa^ dno.rS,uoru.relenb aq1 Jo asec aql
ur "8'a) ado/1od teln8ar e18urs B Jo sacllra^ aql arB
uaqrh
saueld.red.{q g aql
s1u1od >I/urJJT V
a\l
'
'slasor rraql pu"
{rnwr*t} '{r} sdno.r:qns aq1 ' (r't 'Z) ul 'luasarda.r suo8elao 8I stH ',g + g s11ec [g] .d il] uos ,l ,e} teco.rdtcar ot\l aqJ, -ulqou qcrqir uorlsrnSr;uoc aql urroJ ", {e Irac-009 eqr
'( g't'z)roJ
pu" ia,raqds.rad.{q arrr's
r;lt-;:""*
,l "rr ,s} n""-lz 'l'e} teco.iarcar o^r1 ' (v't 'z) .roy: ", {e {e "ur
' (t 't '27 tol 'saueld 1zuo5oq1.ro dlalaldruoc uf u7} o^f ' (u,Z,Z) ",{ rl .rol : .re1n3ar .ro; :sadoltlod .re1n8a.r on1 {uZ}'uo3-u6 "' (r'u'1) ro auo Jo sacrlral aql are s1u1od aq1 as?c r{cea uI .[t ZZ .d ,i ltt t .d -Z] suorlcalJal dq pale;aua8 dno.r8 allur] B Jo ,,sJorJIur I?nlJ-rA,, pu€ sJoJ
-rrru aql ar?
r{crq/n 'saue1d.rad,{{ >{/uru Z Z u1 suol}ca1ya.r dq pa8ueqa.ralul
are slurod ppodrlue ;o stred \/wnV Z aql snqJ, 'I_x - = x - ,q x uorural -enb 11rm qcea saceldar I + saEurqc.ralul qclqa uorlcallar aql leql aaras -qo ol aaeq dla.reu an '.{1.rado.rd on1 due sa8ueqc.ralur
s1q1 qs11qelsa
lsql uollcalJar aql dq
oJ 'sJaqruaur ppodllue
nas11
olq
paurJoJsr.rerl sr
las aloq^! aql l€ql dlrado.rd aW qll/t\'I = xx a.raqds-g aql uo s1u1od;o 1as e dq paluasa.rda.r s1 dno.r8 1e.rpaq,{1od.d"reurq qcea l"ql put} am'aceds-7 uespllcng ur 1ty 'Z*
'I* 'o*; sal€urprooc
uersalre3 q1lin lugod aql dq
[f , p. aSf ], which may be described as foilows"
Consider (in Euciidean 4-space) two 24-cells in reciprocal positions. one so large that
it entirety
encloses the other. Let the Iarge one shrink
(or ttre small one grow) untit the corners of the small one are cut ofi by the bounding hyperplanes of the large one. The part of space that lies inside both polytopes is called a truncation of either. Eventually the large polytope becomes small, and the small one large, so that their roles are interchanged. At certain stages (when the polygonal faces are regular)
the truncation is a uniform polytope. In the notation of "Wythoff's con-
struction" It], ttre complete series of such trunctions of the 24-cell is follows:
as
IL
sE.{rt ,, ,uz} qrnocdauoq oloq,,r arn roJ loq "t ,"2} Erpaqlp arB qcn{.,r\ -ru,{s r1;epcg aler.rdo.rdd€ oql snqJ .[f 'o ,Ol] {Z r) sIIac uZ puE uZ olnlrlsuoc sasual pue sataqdsrwaq 1ea.r3 tuZi ,,saceJ,, 'I
aIqEJ ur
aq,1 'sa8pa uZ olur ,c esodruocap sacrlJaA asaqJ .,a Buole paceds dluaaa 'sac11.raa u6 1r Surar8 dq qruoc,fauoq I"np-FIas E
olul lr a{Bru uE; alA
'[gg '0 'g-] suelpg.rau alaldruoc u dq sarml uZ olul a.raqds ,!.reurp.ro ur ;o ,r} uol1r1red aql sl qcrq/{'{u, uollellassal lecl.raqds aq1 ;o an3opue I"uolsuaurp-g E se pap.re8at aq.{eru a.rnEly
a\I'u/L
el8ue ue 1e Eur
-laaur sa.raqds alllncasuoc '(,c sluasa.rda,r qcrqrrr) olcrrc auo q3no.rq1
Eur
-ssed 1p sa.raqds d.reulp.ro u dq.raqloue auo urorJ pale.redas ar" sasual aql Iapou slr{1 uI 'aceds-g utaprlcng B oluo uorlcafo.rd crqde"r8oa.rals
I"uorsuaurp-t € dq palanrlsuoc oq ppoc s€ qcns Iapou leru.ro;uoc e dq pazrlenst^ dlrsea lsour sr 'eceds-g lecr.raqds aq1 3ur11r1,sasue1 qens uZ ;o qwoc,{auoq aqJ, 'c o1 (,,.re1od,,.ro) puo8oq1.ro,{1e1a1druoc alc.rrc
1ea.r3
aql :,c,,.ro1€nba,, uotuuroc e 8uo1r laaw qclql\ sa.raqdsrruar{ }ea.r8 o4.{q paprmoq suel e sr uorSat lalqclrle aqi pue 'c elc.rqc 1ea.r8 e Buole paards .41ua,re a.rz slurod uA aq1 asec
' ( r 'u 'I)
srql
q
'papnl?xa dlsnorrra.rd alrr qc1r{ilr
dnor8 cr1c,{a ar{l roJ osIE punoJ aq uea uor5a.r B qcns
'[VZ'A
tT]
uolEa-r letuarueprm] € sr IIac aq1 :dno.r8
p.rpaqdlod dteulq aql ]o luaurala r{cea .ro; uor8a.r qcns auo sl araql snql
'sraqlo aq1;o due aq1
;o 1:ed
aq1 yo
01
u"ql 1ulod
s1q1 01
rareau sl leql eceds leer.raqds
Sullsgsuoa'[6I-gIZ'dd'6]
,,uo13o.r laFIclJ!61,, B Jo Jarru?u eq1 ur slurod
-.rns qwoadauoq slql Jo IIer qc"S '[ggI
-q"lqo snql
\/u,ay
louoro1l, ro
? or{1 Jo auo spunor
qruoc.{euoq Iecrraqds e
8u-r
'a.raqds-g Dlrluacuoc B oluo raluac slt urorJ ado1,{1od qcea
Eu11caford dq aceds-g lecr-raqds o1
'", r 't 'ti)--lEnba
1t
e1pplru aql
'd'l]
,,uo.rpaqd1od
acrds-p u€ap-rlang rno"r; ssed all
o/$1 Jo
q sl 'sn su.raruoc dlyelqc
lualuoJ uolnluoc aql :sarras aql
Jo
qclq,r\'{e ,l ,e} z'11 ado6lod aq;
produ"t" {zrr} * {zt , which arises from the } dicyclic group (2, 2, n) , seems at first to be unlike all the other honeycombs. But in fact it can be derived from two reciprocal {Zn,Z, Zn}'s
The "rectangular
in the same manner that we derived,r,,
rl
13, 4,
3
J's.
{t.
+,
l} irom two reciprocal
The appropriate graPh is
which shows that, instead of calling
it {Zn} . {r"}
well have called tt ,1,2 \2n,2,2n1
,
(l
The quaternion
, we could just as
group (2,2,2) has been listed separately (after
the general dicyclic group) because the rectangular product of two squares happens to be the hypercube sociated with the fact
{+,
S,
that <2,2,2> , unlike
S
} . tnt" regularity is as-
the other dicyclic groups,
has an outer automorphism of period 3.
?.
The representation by Clifford translations. When a binary poly-
hedral group is represented by a spherical honeycomb, the cell that represents a quaternion q (belonging to the group) is derived from the cell that represents 1 by a Clifford translation [9, p. t+Z;
!!, p. 138] which
transforms the general point x into xq. This is a left-handed screwdisplacement: the product of an ordinary translation along a line (i.e., a great circle) and a rotation about the same line, with the special condition that the angle of the rotation is equal to the angular measure of the
translation. In the case o{ the cyclic group generated by a single quaternion, the generating clifford translation transforms each lens into the next, with an accompanying rotation. when the rectangular product
{rt} 72
>r
{Zt} t" regarded as a pair
tt
I\9
/\,.\/\,^. Nt w i\9
\9
F
C.,
C.t
C,
t9
19
I
ctra(rt\9lF \,/ \./
v'
v
v
o o t r5
7a i:o .iF
3!0 d'
o
O) t9 O@t\9HAl.9 O@t}O)i'I
Ft
t\,
o
tn
l{
19 Ctr O-fcgqt@19 oo)o)tgF:J
-J
19 O 13 o 5
fi Y d
HF
A(g AI9 or1 o:. f;s
EE to -a
o.
sq
t\,
(o OJ -3. 6
E o a
Nt t5 o '.o tr
o a
t\9 4,5 5I Ne =\e t\3o) p I FA
H a df; Jo aa
Eo
Fi
o a
N9l5t\9@At\9 o@A^lF
o.grotrr5o o-lodE3 otstloF.r 6-lgthgQ
SUdBf +*EsrAj ox: 6EP
t0tr o
a
o o (n
Fl
ri 14
of interlocked rings of stacked prisms
(2,2, n)
lg p. tZ+], the generator
T of
permutes the prisms in each ring, rotating them at the same
time. The case when n = 3 is illustrated in Figure 2, where the edges of a hexagonal prism are marked so as to indicate the manner in which T transforms the bottom hexagon into the top one. The other kind of generator, s (or R), has its axis "horizontal" instead of "vertical," transforming one of the squares (such as 1524) into the opposite one'
I -\\
I
i2 P-- ----\ 4
Figure
Similarly, the generators 24 octahedral cells
or {f,
+,
S and
S}
T
2
of (2, 3, 3)
each permute the
in 4 rings of 6. We can obtain some
idea of the appearance of such a ring by stacking six octahedra (so that
neighbors share a face). With this model before us, we immediately see
certain sets of six oblique edges winding up the column in the manner of a helix. when the top and bottom triangles of the column are brought to-
gether (by rotations through 60o about the planes of the remaining horizontal triangles), these six edges close up to form a skew hexagon which
is shifted along itself by the clifford translation. Figure 3 shows how each face of an octahedron
is transformed into the opposite face by such
a "screw -displacement, " 74
st -tnetr Eurlsorelul u? leql prmoJ [OL-tS '00 TT]
lra]las pue IIBJIarq,l, ,sepod
-11ue Euldyrluepl dq aceds lecg.raqds ruo"ry pa^-lrap aq u?c aceds c11d1ga se
lsnf
'pIoJIuBru l"uo-rsuaurp-aerql E Io clnorb lquarrEpr.mJ aqJ, .g
(',f,
ant
a^Eqeq s.role.reua8 aq1
I1e qclqAt u1 'dno.r8 lBJpaqesocl r{.reu1q aql purJ IIBqs a^\
IIs q
lo suorl"luasa.rd Jeqlo eruos '1! qll/r\ uor.uluoc ur xaua^ B ua^a 1ou Surleq'a1ele
-des d1a;11ua sI 1"ql raqlou€
olq nec qc?a srrrrolsuBrl
'[grz 'a 'zt lcg'g '31,{ 'Ig 'd '7] t-rpaqe-r1a1
S "role.raua8 aq;,)
-re1n5ar paqlrcsul g Suorue
uollnqlrlstp rraql ol Surp.rocce sacllJa^ aq1 5u14retu dq paleclpur
s1
ul {S ,t ,S}
lo
IIac € Io sace; alrsoddo;o 8u1qc1eru oql pue'g1 ;o s8ur.r Erpaqeaapop
:^rr"urr' Jo r rolerauaS aq1
0zI aqt salnruead (s't,'z)
'sa8pa
oql Jo p"alsul sacllral
aq1
Eulqteu dq
allsoddo uaa/r\laq acuepuodsa.rroc aql
pa1ec1pu1 dlalenbape aq 01 saa€J
to; paleclldruoc flluarcryps
sg (p
am8rg) aqnc palecrmrl aqJ 'saceJrolrn puo8elco qll/t\ g ;o s8urt ul tuaql selnurad (g pouad Jo) r roleraua8 eq1 pue 'sace;.re1ur .relnSuerrl r{lllt\ g
;o
p saqnc pal"crmrl gt ar{l salnurrad (g por.rad dno.r8 lerpatlelco d.reurq aql rod Jo) S rolereuaS aq1 '(''t'Z)
s8u1.r
u1
{g'U't}
Z'11
g a.rn81g
z
(, L,
t
,i,B
B
^A-. F Figure
4
ily of 3 -dimensional manifolds can be derived by identifying ali the points related to a given point by any finite group of Clifford translations. The simplest way to do this is to take a fundamental region for the group, as
in Figure 2, 3, or 4, and identify pairs of opposite faces as indicated. In this way the honeycomb is reduced to a "polytope" having only one cell, as in Table
II.
(The numbers of vertices and other elements in Table I
simply have to be divided by the number of cells.) Elliptic space itself arises in this manner from the group (1, 1, honeycomb
rl
72,2,2 |
1)
and the very simple
.
Following Tietze [t+, p. 111], threUall and Seffert describe not only
trom { 2n,2, znl but the more gen< q < p/2 eral lens space (p, q) $rat comes tro* {n, 2,p/q}, wnere L
the simple lens space (2n, 1) derived
and (p, q) = 1. (When g > 1, the gener"ator of the fundamental group is no
longer a Clifford translation.) They give necessary and sufficient conditions for two such spaces (with the same p) to be homeomorphic ?6
[E p. SSt].
LL
\9 q,
/\
t\9
4.. t\9
a\ 19
\9
F
Cr9
q,
N,
\9
F
v
\/
\./
.v,
AA(l,T\9i'I
v
\,,
g,FlOordl' (nFJ
tsl U)o tr Eed EE Eg 8' 's *g[ 3s 3& oo.'9
3.I F!o
I
5F
f3 iE
o H o
o A D')
€
e
o.
o !
o th
Fl F
o
O) r5 6 i' i.f
t\9
t9
Cr,A .)f. llin gE O al'
0c
o a
A ai 3. br E
t' U) F C p
FiJ t\ro 5F I d ocp
; 9B E dY ,dE o a :io o (h
d
rt p.
H t9
I
t
oc
9
t0
o o (n
"'a
HHPPFF
o.footto. of.lcF= nts+*J O;tF:6oo Ao5trO. iiD(iPE :yRo-o
6X!!' 6E9
lo) OF o
C)
o
l.
trl
Along with the spherical dodecahedron space derived
tto* {S, t,t}.
they consider also a hyperbolic dodecahedron space that comes from the
s, l} ltz, p.218; 6, 157]. {0, t_i
hyperbolic honeycomb
9.
Svmmetrical presentations
for
<2,
2,2> and (2, 3, 3) .
We
have so far presented the binary polyhedral groups in terms of three
generators. But
it is quite easy to Eliminate
any one of the generators.
This is particularly desirable when n = 2, so that we can substitute ST
for R, obtaining
9.1
sm = Tn = (sr)z
t)r
9.2
TST =
s*-1,
srs
=
C-1.
r-or instance, the quaternion group <2, 2,2> is simply
9.3
jkj
kjk = 3,
and the binary tetrahedral
9.4
TST =
= k,
group ( 2, 3, 3> is
s2,
srs
= T2.
The substitution S = A, T = B-1 changes the
.r.i
gazg = e,
latter into
aa2A = n.
ri'hence ._; -eA=R_1 _i
^A ^=BAB_1^
_r-
= C. say. We thus obtain e new presentation in terms oi 3 generators:
t.6 i-.etinrng
iB
= AC ..
n = g-iC-1
llA.
ABC =
1.
= C-1e-1 = A-18-1,'we obtain a presentation in
terms ol 4 generators:
!l"t
DCB = CDA = BAD = ABC =
ii:i*: rlo Seiieri and Thretfail [1?, p. 218].
78
1,
flnally. the substitution A = S,
6L
'z=e(JS) =pI=eS uearB t,{les,raluoC
' (V't'Z)
dno.r8 lerparielro.{.reurq aql aur}ap 01 u.,rlou{
ar"
qcrq^\
'r(,r,s)=rJ=ts suorl"Ior
aq1
dldtur (f_X = J pue
z'0r
fI = S r{lln) I.0I suo1lelar .J = J .If .I .II =
aql snql
'VJ = ZL.fJ,I
.fl = ,(rs) = ,(rn) = If 'IJ eS puB Jf = IJ = S acnpap a,n,r_>I = J,fI = g Eururyag ,I=fi>I 'X=IXf 'f =Xfl I'OI suorlelar aq1 .{;sr1es ol uaes dlrsea
U ,2./ 'ET=r ^ FT=x
a.re
,U 'Fi=r suolu
-ralenb aartg aq;, '
(V't'Z)
l"clrlaIIIIII^s .0I
roJ suorlElueserd
'[ol 'd'e]
,-{ ' ut O x (u 'tu '?,)
=
(u ,*- ,Z) ur€}qo at\
u(su;=*S=zu ,{q l'O 8ulce1da"r ',{1.re11rn1g
'9a dno.r8 c11cdc aql oulJap ,{qenlce 'g'6 alqwese.l
'p=cq
'c=e€
,{1p1cgy.radns qc1q,n
,s=pc
'q=ep
suorl"Ier aql
13q1
s{r"rrrar uu?wneN
- }"6 Jo IIv)
('dno.r8 1erpar{"4a1 ,{.reu1q aql Jo suoll€1uasa"rd are g'6
'A=nS 'n=JA 'I=Sn ruroJ sruu?urnsN olur s1q1 slnd
I_A
'S= =
q 'n
J =C
8'6
'I-I
=E
we can define
J=sr-l
I=T-1s,
K=T-1, and deduce
u = t-1s2r-1 - T-1s-1T-12 KU = T-1s = I,
=
s,
K-1 = T,
uK= sK- sr-l
= J,
JKr = sr-3s = srsZ-1 = r-1 = x. Hence 10.1 and 10.2 are equivalent,
i.e., 10.1 is a valid presentation for
the binary octahedral group. Given 10.1, we can equate four expressions
for I to obtain
= 6_1r_1* = J_lKI = KJK-1. 'r_1,1_1 These three equations are not independent but can all be derived from
the two relations
10.3
JIcI = IUK,
.l-ltcl-l
JKI = KJK,
]f,jJ2K = J2.
=
x.l-lx
or
10.4
Hence 10.3 and 10.4 each suffice to define the binary octahedral group. Since the relation
Kl2K
= .12
ir, 1o.e can be replaced by JK2J = K2,
we might be tempted to expect the two relations
r;.2K = J2,
.lx2.l = x2
to suffice. However, the group so defined is obviously infinite, since it has the infinite dihedral group
f=:r<2=t [1 p. 21], where a finite factor x2lx2 = J, J2KJ2 = K is described.)
as a factor group. (Cf.
finite group
The infinite group defined by the single relation
group of the in-
I8
'Sr-r =,-SJS =A
'1_,lS =SJI_S =O
,I_S de pur g .{q I }o srtrro}su"rl aql raplsuoc
'z=zt's)=gr=gs
I'II
dno"rE le.rpaq
-Bsoc1 dreulq aql q . (e ,e ,Z;
'[pSl '0 $] suolu.ralenb
Z/6+ !+t+I+)
1e.r3a1ur
,>f+ .[+
;o walsds s,zg^unH Io
,!+
,I+ sllrm
VZaql 1sn[ e"re dno.r8 1e.rperi"rlal ,{.reurq aql }o
'l*fr--t=c 'a-f,-=a
sluatua;
,1ce; VZ aq1
u1
'n-t',.*r=v puu
olr 'suolu.ra1enb sy
.L.6 dno.r8qns lBrpor{?rla1 d.reulq e aletaue8 snql pu€
'I = I_Xf .I_fI.r_DI = CAV 'I = I-XI_f .ft_I .I}I = (IVA 'I = .,-IX' t-Xt_f .fI = yCC 'I = xr_f .fI . I_II->I = scc d;sr1es qclq.n
,If=xI=tr=t_a 'tr-"=I-Xf = fI=J 'tr-t= I-fI= IX=A
'tr_t= I_IX=)If=V
sluaurala rnoJ aql suleluoc 1.91 dno.r8 l€rpeqelco ,!.rru1q eq;
'[eg 'a
5'i]
toua
llolarl
aq1
;o dno.r8 lelue.o€prml eql se
!
azlell
os1e aas
iggl 'd
dq peuorluaru
)II)I = fXf
s1
g'0I
We find that
uv-1u-lv = s-1ts.s-1r.TS-1.
STs-1 = S-1T4S-1 = T,
vt-1v-lt
=
STS-1.r-1.
s-lt.T
ru-1r-lu
=
T.TS-1 . t-1
.s-lrs
=
St4z-1
=
sr-1=
U,
= T4zS = r-ls = V.
Thus
uv-lu-lv = t,
t:.z
vr-lv-lT = u,
TU-1T-1u = v
and obviously also
VU=TV=UT=S. De Buda has observed that the relations 11.2 suffice to define the
binary icosahedral group. One step in his proof is the deduction of
Vu = TV = UT. This is achieved as follows. The relations 11.2 (beginning with the last) are equivalent to
LL.zL T=uv-1Tu-l, u=vr-luv-l, v=ru-lvr-l or, in terms of three new elements
11.3 A=vr-l, B=TU-l, c=uv-l, 11.31 T=CB, U=AC, V=BA. Thus
11.4
T=BU=BAC, U=CV=CBA, V=AT=ACB; W
and
= ACB.AC = AC .BAC = UT,
similarly these are equal to TV. We have now proved that
11.5
VU = TV = UT,
as desired. In terms of S = VU = TV = UT, we have
82
t8 Zq
= BqI_"qB
'Z? = qEf_q?q
6'I
I
uollrulJap polelor ,{1aso1c aqg,
'VAV = A.VE
'EVE = VaEV
:3 pa^Irap
'I
s1 dno.r8 l"Jpaqesoc1 d.reulq
3u11eu1rug1a
8'I
I
dq 1'11 tuo.ry
eql rol uorleluasa.rd qlrnoJ V
l"ql paao.rd aAEq ad\ ,d11e1uap-rcu1 'aleldruoa sI uolleclJlluapl ar{l snq;,'Z'II salldtur ,u.rn1 u1 ,qclqin IZ.II sa11dru1 IS'II uaq&'l{1.re11ru1s sltiolloJ t.II Jo lsor aql it_IA = v a^sq alrl = CEV d1dru1 2,.11 suorl"Iar aql
'IV=gOV=VA=A
q 't'II pue g'11 d1dru1 'Ig'II rn1,n Suop 'daq1 1eq1 paao.rd aq ol su!"wal fpo U ,(t.II r1llr\ IS'II Sur.redrsoc ttq) g'ff pue Z'II ruorJ suorlBlar esaql acurg ^roIIoJ 'acv = va 'vgc = cY 'cvg = gc ,.'II aculs '1cey
s1
dnor8 Ierpaq?soJl f.reulq or{l ro} uorleluase.rd
prlq
V
'uoI1"lue -sa.rd
plpl dlpnba uB s" Z'II
paqs-rlqqsa
palcnrlsuocar a^sq o^\ snr{J'_(,tS) =
'r_surr_s =
f-sztz(g_srgs)I-s I_SIS' I_S,l'
=
puB'I'II suorlelar Eurul;ap aq1
^S'g'II dq,os1y.gI = tS a?ueqa
,-srrr-srrrs = =I = Z-S.I,ZS r_Sr-IrS
'_SIZS'
pIeId
ol A,-n,_AO = J uI_pelnlllsqns eq u?a srerllo aqJ, 'A pue n q11,n d1le11 t- t-tnls prre 'J qll,n salnruuro? gS l€r{l st\oqs (r-SJ,rS = I) uorl?Ier lsBI eq;,
't-SIgS = r_Sr-JrS = r_AS =,L
'r-sr_rrs=
,-SIS =,-OS =A
.S 'r_sJrs = ,_SIS
=
o ,_,LS = 'rS
=
JSI
9'II
(a = A, b = B-1) has been cited by Neumann as one of the "shortest" pos-
sible presentations of this group. In terms of the "golden section" number
,
=!2{u'U + 1), we have the
quaternions
s
=
| tr-i-:-r.),
t=it, -i-,-1k), u=*t,-k-'-1i), u=it,-i-'-li), o = i t , -i+'-1i), u = a = it"-k+'-1i). The presentation 11.2 shows that the binary icosahedral group
'(,-1+'-1k),
is its
ItS, p. 140]. It seems to be the smallest nonsimple group of this kind. Neumann has conjectured that it may be the
own commutator group
only such group which can be defined by as few as 2 relations (and the
It is natural to ask what
same number of generators).
would happen
if
we
replaced the relations 11.2 by
u-1v-luv
=
T, v-lT-lvr = u, t-lu-lru
=
v.
The rather surprising answer is that the group so defined collapses com-
pletely: its order is 1.'
It is interesting also to observe that the relations
11.5 are equiva-
lent to UVU = VUV (cf. 10.5).
12. A direct product. In the binary icosahedral group
R2=s3=1.5=RST=2, consider the transform of T by R:
w = R-1rR
= Rz
-1:rR
= RS-1 =
r-1s.
We lind that
wt2w = T-lsrs
rw2r
=
r-lRs = T-22 = T3,
=
r.R-1r2R .T
=
z(s-lt)z
=
z-1(rnr)z
= zrtt-2 = w3. 84
98
I_IX'LZZI_XJ,X = ,Zr-IXr_IaIr_XrX zz
z\ _ Lx) z(, _xrr), _,rx
= rz r(r_ rx) zr
_
rx
=
=, zr(, _,rx) pue
z
=
,(r-xrt)
a^sq e& 'X'I_LXZ = XgW = nX =
,XrX
r_re(rx)zx z&zx
=
=
,(xJ,)rxrl.rx
=
I_re(rx)trcXcf
=
,I
aculs ,f,es ,7
=
r_rrJrr = nr acuor{A\
ruorJ srr\ollo}
'r(,r,x)r,rrx = ,I le$ uorl"Iar lsBI stql .Z(JX)rf,rX osIB eculs 1I 'JXTJX = = tJ, tI fIr"IIrEts pu?
xrx' rexzJ = xJX. zxr = XJrXr = rx .X ' ,xrrrx = x erxzxr = XrJrXr = XZI' ,I,'XI'I = X'JX' ZXZL = rJ-rXrJ. = rJrXrJ
,(x,r,)rxr.r,
=
3ugd1du1
'rx sawoceq
I'ZI l"rtl
os 'r_X dq
11
=
J,rxr
=
xrrx
z.zl
eaeldar ol lualualuoc 1l spug aH
'(n = Z 8u1pas
sl
'rr
,{q paallap) dno.r8 lrrpaq"soc-r
Io dno.r8.ro1ceJ I?r4lrl-uou dluo aq1 seareqrr ,g.rap.ro ;o dno.r8 rolce; E s€r{ I'ZI dq paulyap dno.r5 aq1 1eq1 Eulnoqs ,1 = ol
aql
'22
rI
an;r '.rog .dno.r8 leJpaq"socr ^d"reu1q aql aur; lou op I'zI suollBlar aql l"ql pa^rasqo sBr.I acl
ecnpar do,D'r_I = d\las -ap
01 ac-rlJns
"png
'S = lt\J
osle dlsnor,rqo pu€
,At = IZA\J
'rJ
= ,tArJ1tl
t'zt snqr
=
xT6x-1Tx6T-1 = (xr3x-1)2(rfr-1)2
= $2x-2rzrx2r-212 =
1.
Interchanging T and X, we have similarly (rx-1)323 = 1. Thus
z Defining
S=
=
(xr-\-3 = 11;-1;3 = z-3.
TX-1, so that (TS)2 = (r21-1rZ = Z, we have
15=(sr)z=g,
53 = 23,
The further substitution T =
26=L.
t
T Z' yields T!,5
= z-9
-3. , (st')z = z
Thus
another presentation for the same group is s3 =
r'5 = lsy'12 = g3,
sz =
zs,
TZ =
zr.
1st')2 = z' impry z'2 = 1, it is no longer necessary to mention Z6 = L.) This is the group that was denoted elsewhere by <2, g, 5> [9, pp. ?0-?1]. It is clearly the direct product of (since the relations s3 =
1'5
=
t-}
cvclic s"oun {22 Thus 12.1 and 12.2 are symmetrical presentations for the group
the binary icosahedral gto.tp
(2, 3, 5) x
{s,
ana the
}
.
G3,
of order 360.
References 0.
S. A.
Amitsur, Finite subgroups of division rings, Trans. Amer.
1.
$;i;
Y;f,l;:',ftk
2.
Math.soc.So(lffi
'
ralizations
- The binarv polvhedr
ot the quaternlon g!o!rp, ljuKe
, Quaternions and reflections, Amer. Math. Monthly, 53(1946),
!..
Fp.T36-j+e
.
t
Regular PolytoPes. London, 1948.
q.
Extreme forms, Canad. J. Math. 3(1951), pp. 391-441.
6.
-
rboli
-, 86
Proc. Internat.
,.8
'rua.o..'oc
tl,:8"'.T".9 '9I
.Rrr-r
.aa ,(sonr)ET .sdqa .qren .r{Sreugnr ,Edrrd-{FrJle75!ffi8fr JaIBUO!suaIIIrprqau ualu"-rr"Arll uaqcsrboloctol arp raqn .azlerl .H
'rl
'dd lg
saurn?l{ uaqcs cllpue aqc!aJ
rap
IbOI
'7961 '31zdya1
,
'1.ray1ag
'ffi'1e;1a.rq;
uaur
rnurluo{s!q
'H pue II"JIarqJ '111
pue
lra]las
a
rap uollcnpau alp
uaqcs!
.H
'zI
'qlell'f
'j
larp llw uarrrJoJ
,, '1a1qcr.rrq'1
'g
'0r
'6
'(' J - N)'q1e1a1'qa8.r g'Fil[-o-.r-5-5]5"raffi 's 'H '8 'rasolN 'r 'o '/I\ puB ralaxoc
'n
JoJ suollBlar p
'dd'(SSOf)f
'tr
,Jlx{,;18?;0"{$ .II
'gI6I 'uopyo.I '(tggl 'SIzdlaT ' ua8unselro1 Io uorlelsrre.r;) uo.rpaqesoJr ar{l uo Sarntcaf 'ulaDl 'ina3ue
'71t
'peueC
ffi
,___El:gl
.l
APPLICATIONS OF GROUP CHARACTERS by Michio Suzuki
We shall describe a method of applying the theory of group charac-
ters to a partlcular type of problem in the theory of finite groups. The particular type of problem which we are interested in is the following one: To what extent and how does the knowledge ol a subgroup H deter-
mine the structure of a finite group G which contains
H ? The
theory of
group characters provides a method to attack this line of investigation. Suppose that we are in the most favorable situation and know everything
concerning the subgroup H. Then we can determine
all the irreducible
characters of H. From the knowledge of these characters of H we can get information on irreducible characters and finally on the structure of G. Our main concern is how to get this information. Essentially, this is done by applying the Frobenius
reciprocity law. We shall however give
a
modified form tn the second section, which may be more convenient ln
particular cases. The first section is devoted to the discussion of basic material necessary for the later development, and in the last two sections we shall give a few applications of our method, including a proof of the following theorem recently proved by R. Brauer and myself: a 2Sylow subgroup of a simple group of even order cannot be a generalized quaternion group.
1.
We conslder a
finite group G of order g. Let X!, .
.. ,
X
n
be the
totality of distinct irreducible characters. We shall restrict our consideration to the classical case of Frobenius characters so that the number of distinct irreducible charactere is equal to the number of distfnct conjugate classes of G. A (generalized) character 0 is a linear combination 88
68
ra5u"p ou sr araql Jl (C)W dq,{1dru1s .ro'(C)DW,{q palouap sr O uo qsrut^ qclqa DIAI Jo sluaurala yo,{111e1o1 aqJ'D u} C Jo luerualduoc aq1 3 dq alou Jo auo yo u1oI r s1 -ap ar! tC las pasolc qc"a JoJ 'sass"Ic q?ns IEJaAas
pall"c st C Jo C lasqns V 'sassrlr acualzalnba olur sluauala dno.r8 ;o uor111.red sacnpu! aloqt paurJap uorlelar acualennba aqtr,
C JI pasola
"
pue (ur poLtr)
t
.I = (u,)t)
=
{
leql qcns
{
la8a1u1 u€ sr araql uaql
'u/ru pu€ I
=
ar€ u'rtt'T JI l€ql slrass? qclr{a\ €ur -rual lBcllaurwlr" aq? pasn a^"q a,t;oo"rd aq1 ;o lted rallBl aql uI'H Jo rol -e.rauaE e o1 a1e8n[uoc sr , lsr{l suBaur qJlr{ltt''0 #(t) *0 al€q a& acuag (r;u'tr)
'*0
y1
pue s.ra8a1ur I€uorlBJ aarr{l
Jo uollrurJap eq1 o1
d;e;1uoc 0 = *, lEql su"au
slql 'I
= (3,{) qll,tr )t
)*B aruaH'0 = (d)*d uolllurJap,tquoql .0 =(o) *g ;r pu" d o1 luapllnba src II 'H Jo .role.raue8 auos 01 e1e8n[uoc 1ou sl r Jl 0 = (") *0 raaad\oH '0 f *0 pue DW Jo luawela u" sr d urorJ paonpu! *A relc"reqc eq;,'H Jo / rolerauaS-uou due .ro; 0 = (r) d pue 1 fi1c11dlllnu qlra H Jo relc"r"qc l"I^lrl aql sur€luoc d 1"r{l qcns HW ul d luaurala u€ sr araql uaqJ' d,{q pale.raua3 g i, } = dno.r3 -qns aql Japlsuoc eA! asJaluoc aq1 aao.rd ol rapro uI 'd 01 lualennbe sr o uaql'I = (3')t) qlle {d o1 ale8nluoc s! , ]l 1"q1 snopqo d1.r1ey s1 11 IIB JoJ0 = (oa
'd -uoc
ol ruarearnba s1
,
Jr
sI o p fluo pu" JI d
8uytro11o; aql alrel{
acuaH'I o1
=
(s';ril:t::t;:"::::;'"t;r:r:::
lualealnba
su
luawala
uy
an dlaueu lasec aql ralat\oq sI sIqI 'uoll"Iar acual
-earnba u" saurlap,{11rn1ce uorl"Iar aloq" aql
0 = (d
)/
sagldrul
y,g
r
p
l"r0
snorAqo lou
roJ 0 =
sluaurala o^\1 IIBo
atl'u {uer yo (dnor8 u€Ilaq€
l"ql rtalc s! u 'O Jo sralc"rtqc I=I 'u'3
=
d
yo d111t1o1
(,
sl
1I
)p
tuatei\in65 C
Io
aa.ry) blnporu E
sI
lf
pu",
'1 rua.roaqg,
aql alouap Oy{ =
d IAI
w 1"T 'l x I"
:slualclJlaoc 1rr5a1u1 l?uoll"r qlrd\ sralc€r€qc alqrcnporJ!
Jo
of confusion. Clearly the set M(C) is a submodule of M which is in fact a direct summand of M.
Lemma
1.
There
exists , e M(C) such that d takes a constant
value c on C, where c is a rational integer dividing the group order g.
This follows from the fact
that laeC x (a) is rational for any ir-
reducible character X of G and
foi
any closed set C.
As a corollary we get Lemma
2. If C, and Crare disjoint
closed sets of G, then the index
tM(c1 U Cr):M(Cr) u M(Cz)l is finite. Let r(M(C)) be the rank of M(C). Then from Lemma 2 we get
r(M(C1u . . . . . ucs)) = x i 3 1r(rvr(Ci)) if c1,,
cs are mutually
disjoint closed sets of G. Theorem
2. Let C be a closed set. If C consists
classes, then r(M(C) ) =
of
t conjugate
1.
Let 0y. .. , 0r be linearly independent elements of M(C). Then each 0 i is a class function taking values in the field F of the g-th roots of unity. Suppose we have a relation 2a, 0 i = 0 with a, e F. Taking a
base
rationals we can express di as linear combina-
{.0} ot F over tions :k aik ,k with rational
Let X be an irreducible character of G. If aO is the multiplicity of X in t , I t aro d rr then loi di= f ( Xa,O 0 i),ycontatns Xwiththemultiplicity I"k,k. 0. Since allthe Since t i1 r "i di= 0, we conclude that xk coefficients atO.
"k'k=
ak are rational numbers, we must have aO - 0. SinCe X was any irreduc-
ible characterr rve conclude that
!i I t *it di = 0. From the definition,
the d, are linearly independent over rationals, whence aik = 0 for all i 0rare andk.Thereforeweget oi=0 (i= 1r2,...,r)'Thus d1', linearly independent over F andhence r < t. If C1, . . , C" are the totality of closed sets of G which cannot be expressed as a join of two dis90
I6
*/' "'rlp
uolleulquoc r€aull
1o
sluawala luepuadapul
" "l
f1.reaug1
ld'' " "'l
ul
I are oraql 'g ;o sasselc
slslsuor (I II 'H ]o o las posolc -.r1
Id qces'(q)Hw
raprsuoc 'H
a1e3n[uoc 1 ;o
sralcsr"qc alqlcnpar
]o " ' rlf tat'gyodno.rsqns"aqHlaT
pd111e1olaWaq*4 ''
0
'(9)w
'Z
= (c)Ur tew aas o1 fsza
lr'((O)W)r = ((O)IS)r snqJ,'l - u = ((C)w).r - u F ((C)rS)r aauaH '0 = (C)W U (C)r,JpuBqraqloaqruO't - u =((g)r,r).r < ((C)rJI)ruaqJ, 'sass"Ic aqeEnfuoc I Jo lslsuoc C laT '(C)ry;o 1.red E sl (0)W d1rea13 &ou sr
'(O)w = 131t9 '8 tuaroaqr
'(C)w' f ue roJ0= C
<
,1.
' f, >
1"q1 qans W Jo
4
slua{uala;o
d111t1o1
oql's! trrn'(C)W
Jo
an'3 1as pasolc fue.rog .!ql'I=!3=C<,l,rf >
luarualdruoa 1euo6oq1.ro aql (C)Id dq alouap
p1ald suoglela.r d11puo3oqlro aql uaql
!x !q 3 =4, putlx !" g
=+
}I '0 = + ]l fluo pu" }I sploq o
<
(
f
)Crn
aql pu"
( /)D,n
uaql 'C < -ap sl
d111enba
#'
+>
s"
paulJ
u 'Cyt ra^o l?uolleun; rBeuIM z sg lcnpo.rd rauul slql uaql
'(r-.)i (')f c"3 @/tl = c dq lcnpo.rd Jeuul uB aulJap an Ctrq
'z
"ruuarl
u1
urorl srl\ollo} as"c le.rauaE eqJ, 'slas pasola 1untu!
-ulru aql JoJ Z r.ualoaq;, salo.rd s1q; 'u51s
d111enba
aql
aABI{
lsnru
arrr acual{
.u=Il 3 > ((!c)w)r< 1aB
'((!c)rnr)r!< =(("cn
a.r tsasselc ale8nfuoc ]1 yo slslsuoc IC
n I3)nf)r
=u
;1
ua{l 's1as pasolc dldua-uou 1u1o[
with integral coefficients: o
L=
2flr "'.
dr..
Let A be the matrix (arO) of those coefficients. The induced characters 0
|
are elements of M6i let the decomposition be
,l= . b. x I >n ts=r rF I The coefficient matrix g =
(blr ) is an integral matrlx. If we know the matrices A and B, then the values of the irreducible characters Xp on the set D can be obtained. In fact we have the following theorem.
4. Let X = (xO u ) be a solution of the matrix equatlon B = AX with integer" *k, Then for any element p of D we have Theorem
xn(n)=:[rxon
ds( r).
In order to prove this result consider the.element
x= xplH- t[r*ku where
d1,
Xp lH indicates the restriction
of Xn to H. Clearly
that f . Mrf(D). By Theorem that ,i . f4H(D). We have
We want to show
the proposition
3, this
,,l,
.
MH.
is equivalent to
<*, oi>H= .*nlH- :[r ttu 6k, oi> H = <x nln, ei >H - : [t *r." < d6, di> g. By the Frobenius reciprocity law we get
.*nlH, di >H= .Xp,
di*t G=birr. 0' since 01,'', Hence <*, 0i > H = bi* :[t "il.*k,, = iinearly independent, the above equations imply that P . ftH(D) Corollary. the
Suppose
values x* (n) for
grare .
that { r r, . , 0 t} i" basis of Mn(D). " p
€
Then
D are completely determinedby the matrices
A and B. In fact we can find two unimodular matrices S and T such that SAT 92
86
-sod Jo raqrrnu allul; 1r.r5a1u1
dluo a.re areql pu? (g'og) ur.roy eql s€q g ,x1t1eru
"
u" sI g aculs 'tr
Jo
xlrl€tu pasodsue.rl aql sal"alpu! Ll araq,r
g.g 'v.v {t = uorlrnba xuletu aql sp1a1d
suoll"Iar dlllruo8oqpo
qlld\ paulquoc
aq1
a^oq€ ar{J,
"urtual ' 4 > 1a3 ain,(q)w,
'uoluass" rno pu" 0 =H < f / rg aculg'(C)Httt ,4 aJaqAt i + 0 = Hl*d tzqt aas ad\ eurrual dg
'H.g'Hl*at 1eq1 sa11dtu1 ,ne1
=C
a*,!r'*0>
firco.rdgcar snluaqord aql lcEJ uI
'H.+r0,
=C<*f ,*d>
uaql'1q;Hry ,
9 '6i II 'tEruwa-I
aao.rd arn
'gt d
.roy
(d
s1q1 Eu1s61
"uuroT
)d = (d )*, uoqt'(C)H4 ,p JI .t "ruua-I a^?q aA\ (z) pue
-s" aql rapun l"ql s^\oqs sralc"reqc 'H uI paur"luoc sI D ul
pacnpur aql Io uollelndruoc aql
d Jo Jazll"rluac aq1 ueql '6 r 6 il
ale8nfuoc a.re
u1 a1e3n[uoc
ar€
6J
Jo
uaql
(Z)
tg
ur
sluauala o,[l
11
pue
daql uaql rg
(I) suollduns
(I) :suoIl
-lpuoc leuolllppu on1
uoll"Ial -uroJ dJaA
aruos pulJ
"
3u1rno11o]
utc
ad\ asec
aql satJsll"s g les eql l€r{1 asoddng
'g pu" v
uaa&laq
ltrcads e ug'.raaa,tog'rauueur
pa1ec11d
ul C Jo arnlcnJls aql uo spuadap g xrrltur
'xy = s uollenba
aq1 le.raua8 u1
ar{l Jo uollnlos € s}
("or)
xlrl"u
l
=
eql uaqJ '1 aa.rSap Jo xrJl"ru llun aql s!
x
I araqar (O'f) =
sibilities for Bo if the matrix A is known. A particular case of Lemma 4 gives the relation wO(
for
any
de Mg(D).
ff d is a linear
A
*) = wH(
combination of two or three
d)
irre-
ducible characters with multiplicities 11, then d* has a similar form.
3. If the order of a finite group is even we have a method to investigate the structure, which is not available otherwise and which has been
proved to be quite powerful in many cases. This method was
first
ap-
plied by L. Red6i in 1950 and shortly afterwards, independently, by the author and by R. Brauer and K. A. Fowler. The method depends on the
If z and r are two involutions of a finite group G, , then the subgroup {, , } Senerated by the elements r and z is a dihedral group. Let p be the product r n of r and r. Then we have a Pr -t '= p -t '. Hence the involutions r and z are contained in the
following remark.
normalizer of the subgro"p
{. n
senerated'by p .
}
Brauer and Fowler have stated the above idea in the language of group rings. Let T be the sum of all the involutions of G in the group
ring of G over a field. The sum K of elements in a conjugate class of
is in the center of the group ring and the elements K1,
.
G
,Kn corre-
sponding to n distinct classes form a basis of the center. In particular
t=Kt+...+K, where Now
Kl, . . . K, are all the conjugate classes consisting of involutions. ,
T'can
be expressed as a llnear combination of K1,
. . , , Kn with
non-negative integral coefficients:
^2 = -n Ki. Tx i=r "i The coefficient c, indicates how often a fixed element p of K, can be expressed as products of involutions
r and o . This number coincides
with or is one less t}ran the number of involutions in G which translorm P into its inverse o -1. Th.r" if we know the structure of the normalizer 94
96
aql qlFr paurquoc a^oq€ paqrJrsap poqlaur pue In7 v'n[v
aqtr uaqJ, 'i{1aa11cadsa
"
zn
Tv
aq N ul " puzl t ' ! Jo srazll€rluac aq1 Jo srepro aql laT 'r{ }o rosr^rp sl r uaqJ, ' T z aq s }o razllErluac aql Jo rapro aql laT " '(Z poru) I = q 'qg aq N Jo rapro ar{l lal pu" t, uollnlonur aql Jo N raztlerl -uac aql raprsuoC'C ur raqlo qcta 01 ale8nfuoc aJ? r pu? Z, ,l r leql Z
luatualels aql ol lualelrnba sg Z xapul ;o dno.rSqns I€rurou
acualslxa
" ]o Z, = luawala
sr ul " ,"} 'Ir t_o = ul",{t,
-uou aql uo uorldtunss" aq;, 'uollnlolu! uE osIE
=t ;I,
-ur raqlou€ pue
sl
r
Zu
r
aqJ
uorlnloa
dq pale;aua8 sg g dno.r8 aqtr, 'S Jo
luawala
ar{1
l€ql qcns'? JapJo Jo
r
uotlnlolul l"rluac aql Juaurala u" sur"luoc S.Z
xapu! Jo dno.rSqns Isturou ou suleluoJ C 1eql pu€ 'g rapro ]o dno.r8 p.rpaq
-Ip
"
st s lBql aulnss€ a/t\ 'g ;o dno.rSqns noldg-z
;o dno.rE
larl 'rapro ua^a allulJ B eq D 1e.I 'asec lercads E raplsuoa ain alduexa ue sy
"
aq s
'pasn aq u?c sralaEreqa Io anl€ll aql alndruoc ol uorlcas snorla.rd at11 ut paqtrcsap poqlaw aql uorlcauuoc slql uI 'E .rap.ro dno,r5 aql roJ EInuroJ saa.rEep
tu,nou4
""e
r
1aB o1 alqrssod
snolrtA Euowe uorlelar
"
sl
1I
uallo 'F pue ( d x )tA
saqsllqBlsa uoglrnba a^oqt aql uaql
!a pu" sralc?r"qa alqlcnparl
II"
roJ ( d ;
/
1
sanlea aq1
'r
JO
11
Jazll"Jluac
arfi Jo rapro aq1 Eulaq ( r )u pu" suorlnlolu! ;o 3u11s1suoc sass"Ic a1e8n[uoc
uro"rl
|r,
{ }
saAll"luosaJda.r yo las E
ralo uaqel 3u1aq araq uorJer,uuns
,(')u/(,;/x'3
aq}
=1d1;Ca
pur's.ra1ce.r€qc alqrcnparJl tC dno.rE eql ]o rapro aq1 s1 3 araqa aql II€ raao saEus.r uoll"ruruns aql
(rx)Bq/(d)dxr{dx)Dr al€q ad\
dK g=Ic
(*)
'!r
uorssa.rdxa ue p1a1d sralcerBqc dno.r8 ;o suollelar "o1 d111euoEoq1.ro aql puer{ raqlo aql uo 'lc anl€^ aql &ou{ a& u"qr ' lo
{, }
computation method in the precedihg section enables us to obtain the fol-
lowing result. The order g of G has the form g = o4hu2(ut*,rr)2i{i
+, )/(t -,
)2
f is the degree of an irreductble character X and f = e (mod 8) with e = * 1. Actually we can say a little more. The group G has 5 distinguished irreducible characters Xo,X1, , . , ,X4, of which Xo is the principal (trivial) character and X, is the character X mentioned before. Let the degree of Xt be ft: fo = 1, f1 = f. Among those degrees we have where
two linear relations:
f4=f* e=fZ+ €'f3 where e ' is either 1or -1. As a matter of fact the equations
x4(p) = x( p) + . - x2( P)1 .' X3(r) are true for all 2-regular elements p of G. For the 2-singular elements the values of those characters can be computed. Putting
3-
e
e', we get
X1 = e 'e
x'=
t
-e ' x3 = 6 , -5
x4= o'-2'' first number in each row stands for the value on elements of order divisible by 4 and the second one for the value on
In the above equations the
elements of order divisible by exactly 2. The degrees are connected by another relation: g = 256uh2lt
rfr/l (t+.)(fr-.)(fB-
s ).
If 6 = -1, then it is easy to show that rL2-_+t3-_(f.+,)/2. |
Moreover we have the
equality.l
= u2.
If 6 = 1, then those
equalities
are in general not true. As examples we mention the groups LF(2,q) with
1,6
H q d 'n+> ((^+)Ba/.('+ D
r
. *o ' xu
1a3
-sn'0=H < 0',1,>
an
aABr{adr(C)Ey,f
/r( Q)Ht') '< q-(Hl ^
"*
)((
/i a
(*,r)
,ne1 r{11co.rdrcar snluaqo.rg aq1 Bu1
,B
due.ro1 aruaH.CluosoqsluBA ^
d
)Hl) '.q=
/ 11'*)Bitz1 1;c1;
Q(('
tx)cl) x)Bcr/z(
+ )8o.
d3 3 =
uolssa.rdxa aW
,l
l"ql su"au slq;.
'H Jo sJalc"reqc alqrcnparrl aql II€ JaAo saSue.r uolJeruuns aql alaqa\
'(^f )ao11a1"4 r{'+)H,r, niq=t" 'd
.roJ anIEA aurgs aql seq (1) o1 .re11ru1s H roJ
"InruroJ
aql acuaH
.I_d olut d ruro} -sutJl r{clq& c Jo suo}lnlo^ul }o raqunu aql s! 1 + lc .to lc uaql'lc .ro; .roJ (i.) BlnruroJ aql allrlrr a,t tl1 uI sI g Jo d luelua1e ue Bursoddng 'H ul ar" suorlnlolul r{cns IIE (g) uol11puoc aql dg
sI r uaql JlpuBq rd J!(z) 'H uI
'cJo, uollnlo^uraluosroJl_d =l_tdt fg;o
lasqns pasolc
"
sl C (I)
:suolllpuoc 3urno11o; aq1,{yslqes
qclq,[ H ]o q lasqns E pu" D 1o g dno.rSqns B s! araql lEr.Il aulnssv 'd11nc1yg1p
ran"rg
Jo
str{l Jo p1.r
1a3 01 Z
uollros }o poqlatu aql pu" rald,od pu"
poqlau aql aulquoa II€qs adyalnduoc ol llnclJJrp are sral
'lx uI d luaruala aql uo (a ) d x sanl"A sJalc"r"qc Jo sural ur !c Sulssa.rdxa (*) elnurol aq;, 'V
-c"r€qc Jo sonl€^ esaql dllensn aql sa,rlolur
'aJar{ passncsrp aq lou Jaaall\oq
seq g dno"r8qns
lltfi\ qatq&'sl1nsa.r J"Irurs ur"lqo ar\ pu".rap.ro.ra8.re1 rnoldg-g arg uaqa sasec uI s{JoA! IIIIs poqlaru rng
'tt
zt
1uo dno.r3 3ur1eu.ra11r aq;, 'Zn = In pu" = ZJ sdenle aa€q ar\ as€c stql uI '(gI potu) r F b lnq (g poru) r = b
1tq1 aldruexa
tj
f
u" sapllord
s.ra11a1
forany B.Mr(D). If the subgroup H and the subset D satisfy the assumptions (1) and (2) of the section 2 we can apply the results of that section. In particular
if
MH(D) has a character g with
smallwt( a),
then w6( d*)
is also
small. Hence in the above formula the summations on both sides contain only a small number of terms. We shall therefore get a relation between characters of G and those ol H, which is fairly easy tohandle. As an illustration we consider a special case. Let G be a group of even order and let s be one of
its 2-sylow subgroups. we assume that s
is a generalized quaternion group of order
clic subgroup z
of.
2
16. Then S contains a cy-
index 2. since the order of Z is at least equal lo 8, z
contains a unique subgroup P of order 4. Let H be the normalizer of P
in G. It is not too difficult to show that H contains a normal subgroup M
suchthatM
n
S= eandMS= H. LetUbethe subgroupof H suchthat
V -- MZ. S contains only one involution which we denote group V generated by M and
r is a normal subgroup
by '.
The sub-
of H. Let D be the
set of elements of U not in V. It is easy to verify all the conditions of this and the second sections Since
degree 2.
for H and D.
H/M 3 S, all non-Iinear irreducible characters otH/M are ol
Let
d be one of the non-linear irreducible characters of H,/V.
We have at least one such character since H,/V
induced character from a linear character
outside of U.
Let I o be the principal
trivial character ot H/V.
of"
) 8' { is an
U/Y and hence vanishes
character of H and 'l 1 the non-
Then the character
0 = no+ ,1 1-
is of order
d
defined by
Q
w"( d) = 3, we have wO( ar) = 3. Hence d* is a linear combination of three irreducible characters of G with multiplicities + 1. One of those characters must be the principal character 1 of G and
belongs to MH(D). Since
the multiplicity is 1. Hence we have 98
66
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ON MAXIMAL SUBGROUPS by W. E. Deskins
Consideration of the question, What do certain intrinsic properties
of a maximal subgroup H of a finite group G imply about
G
? has led to a
number of interesting results. For example, G is a solvable group
if
(i)
H is abelian (Herstein4), (ii) H is nilpotent with regular (in the sense of
P. Hall) Sylow subgroups (HuppertS), (iii) H is nilpotent with Sylow subgroups of class 2 (Deskinsl),
or (iv) H is nilpotent
and of odd order
lTho*pson6). So one might expect G to have some nice properties when H is supersolvable. Following this line of thought we prove in section 3 that when H and certain other subgroups of G are supersolvable, G is solvable (3.3). In proving this another variety of problem involving maximal subgroups was encountered, the investigation of which should be useful in handling the question mentioned above.
If
.1 is afamily of maximal sub-
groups M of G determined by some "external" relationship between M
is the nature of. S(1), the intersection of all the Mof. 7? These generalized Frattini subgroups will be considered for several different 7's, families selected according to the index of M or variations and G, what
thereon.
family I is through consideration of the indices of the maximal subgroups of G. So, for a fixed
1. Index. A rather
natural way to select a
rational prime p define g = 3p to be the set of all those maximal subgroups M of G with (p,[G:M]) = 1,[G:MJ denoting the index of M in G.
is a metanilpotent normal subgroup of G' 1.1. 6 (7 pp ^) = d '. (A metanilpotent group is an extension of a nilpotent group by a nil100
IOI
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put
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1ua1od
2.1. The number n = o(H/K) is unique for M. Moreover. if [G:M] is a power of a prime. then H and K are also unique.
This may be proved by using induction on o(G) and by investigating the centralizer ol H in G. Hereafter n will be referred to as the normal index of M in G. We
list
some results concerning n.
2.2. G is solvable if
and only
il
every maximal subgroup of G has
prime power normal index.
2.3. The intersection of those maximal subgroups of G whose normal indices are composite is precisely the maximal normal solvable subgroup of G.
2.4. The intersection J of those maximal subgroups of G whose normal indices are prime powers is a subgroup with the property that the socleof
G/Jis@.
2.5. The index is iJ and only
if
a
divisor of the normal iqdq4,4t4! G is solvable
the index and the normal index are equal
for each maximal
subgroup of G.
To prove the sufficiency of the last statement, proceed by induction
If G is simple then the identity element forms the only maximal subgroup of G, so G is abelian. So let L be a minimal normal subgroup
on o(G).
of G; then G/L is solvable by the induction hypothesis' Let p be a prime
divisor of o(f,). Then any maximal subgroup of G which does not contain L has index divisible by p. Therefore L 9 do and hence is solvable. so G
is
an extension of a solvable group by a solvable group and hence is
solvable itself. Proceeding in a direction suggested by one of the characterizations of the hypercenter, we consider the family of all those maximal subgroups for which the previously definedg/r is not in Z(G/K), the center
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.r1ed
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,c
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{w,r} , "qf {w,C}
(q
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'C =,IAI pue 3 dqpalz.raua8 1"ql qcns g
aq O
go dno.r8qns E aq C
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Jo
needs careful scrutiny
is ttre one arising
when G has a unique minimal
normal subgroupKrvithK n H= 1. So let Mbe amaximal subgroup of G not containing K, Then M
is supersolvable,
and we shall show that M
is conjugate to H. As a supersolvable group, M is an extension of a nilpotent group by an abelian group, as is H. So, if H is not abelia.n, there exists a prime p such that both H and M contain normal p-subgroups. Furthermore, the
following simple lemma shows that a p-Sylow subgroup S of H is a pSylow subgroup of G.
Lemma. If P is a p-group expressible as PtP, with P, normal in P and
P, 0
P2 = 1, then Z(PZ)
n
Z@)
+ l.
Therefore M contains a p-Sylow subgroup St of G, and the centers
Z ardZ, of S and Srr resPectively, are normal subgroups of H and M, respectively. So there exists x e G such that x Z *-L = Zl, and consideration of the normalizers of.Z and Zt leads to the conclusion that xH
x-l= u. Then,
if q is a prime divisor of [G:H], it follows simply that K 9
dO,
which is metanilpotent by 1.1. Therefore G is an extension of a solvable group K by a solvable group H and hence
is itself solvable.
References 1.
Deskins. W. E.. An extension of a theorem of Herstein, Abstract 548-
2.
Gaschiitz, W.,' Ueber die d -Gruppe der endlicher Gruppen, Math. z e-iilT3_(1 e 5 3I-IforI?O.
HalI, M., The Theory of Groups, New York, 1959. 4. Herstein. I. N.. A remark on finite groups, Proc. Amer. Math. Soc.
3.
ffi
5. Huppert,B.,Nor@
6.
Thompson, J., Proof of a
endlicher
conjectut 104
, these Proceedings.
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1 e.rqa81e alT reaull aql urroJ uaql a^\ ,{1pcg1uap1
qslue^ lou saop (1) ru.roy aql qclrl^\ uo V Jo S lasqns € uror]
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(r)
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'p
nn automorphism of
L/Lo which we denote Uy 0 g. Then g -+- d, is a
homomorphism of the given linear group G into the automorphism group
Aof.L/Lo,
and the structure of A, at least in principle, is known from
what was said above.(3)
Ho*"u"r, in general it may
happen
that
d
(G) is
"too small," so that the knowledge obtained does not give sufficient insight. Significant results may be expected
if
d
(G) can be proved to be
the restriction to F of a certain algebraic subgroup of A. For this purpose the choice of the set S has to be made in a special way.
To give an example let us assume that G is generated by a set S of
*rt"i"""(4) ("eF)
exp("X)
(2)
where each X satisfies
(3)
XP = 0.
In this event the matrices (2), for any fixed X, form a subgroup U of exponent p of G that is the restriction to F of a one parameter algebraic
e.(5) wtr"r, can we conclude that the group G that is generated by the re-
strictions to F of the algebraic groups gAg-1 (g . G) is the restriction to F of the algebraic group B generated by all gAg-1? This problem in general is not solved. However, let us make the additional assumption that for any matrix
in (2) we have (4)
for Ye
xpYxp-ts
=0
L, t'=1,2,,.., p-1i (4) isfor
instancesatisfied if f S(p +l)/2.
In this case we have the identity exp (o X)Y[exp where X(Y) = XY
(" x)]-1 = exp (" x)(Y)
- YX. It follows that the Lie algebra generated over
by the matrices X is transformed by exp ( " X) into itself, so that this 106
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108
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TITS, J., 38 Trace bilinear form, Transfer map, 1?
Quadruply transitive group, 3?
Quaternion, 65 group, ?8
Translations, 43 Trefoil knot, 81 Truncation, ?0 TVisted field planes, 49
Radical of a form, 104 Regular S -group, 18 polytope, 69 subgroup, 22
Unit quaternion,
SCHLEGEL, V., 6? SEIFERT, H., ?5, 78
Semi-nuclear rings, 49 Singer's theorem, 52 Solvable group, 150 Spherical dodecahedron space, Spherical honeycomb, 70
S-ring, 30 Steiner triple system, 40 suzuKI, M., 25 Tetrahedral group, 65 THRELFALL, W., 75,78 TIETZE, H. ?6
104
6?
Veblen-Wedderburn planes, 48
systems,
78
VINCENT, G., VORONOI, G.,
45 87 ?O
WALL, G.8.,24 WIELANDT, H., 33, 39 WYTHOFF, W. A., ?0 ZASSENHAUS, H., 25
Zero dimensional group,
110
56