Finite groups in which the centralizer of an involution is ·0

JOURNhL OF ALGESRA 32, 173-177(1974)

inite

roups in which the Centralizer LARRY

of an ~~~~~~ti~~ is 1

FZNKELSTEIN

Tne main object of this paper is to prove the following: THEORERI.

N =

@c(z>

G =

O(GjW.

Let

G

be a jinite

is isomorphic

z an inoolutiott

group,

to the group

. 0 disomered

0~’ G and b!

J. Cmzcay.

sup$~se Thn

Our method of proof is fairly straightforward, namely tc first show that H contains a Sylow 2 subgroup of G and then use fusion analysis to verify that z is isolated in H. Glauberman’s Z* theorem [4] may rhen be applied to complete the proof. Notation will follow [5]. In addition S&(G) will denote the collection of Sylow p subgroups of G and B,(G) will denote the collection of all elementar! abelian subgroups of order 2” of G. I.

PROPERTIES

OF .Q

XND

ASSUMED

RESULTS

The general reference for the following discussion will be [2] in t\.hich Conway first constructed 0 and described some of its larger subgroups, Given a set Q of 24 elements we turn the set P(Q) of all subsets of Q into a 24-dimensional vector space over F2 bl- defining the vector addition of tw5 elements of P(Qj to be their symmetric difference. Let g be the subspace of P(Q) spanned by the 759 S-element subsets, called special octads, which form the blocks of a Steiner system S(S, 8; 24) on 52. Then %Thas dimension 12 and consists of the emptv- set Z, si), 759 speciai octads and their complements and 2576 12-element subsets called dodecads. 1TTedenote by F8 , FTre , and W,, the sets of special octads, dodecads, and complements of special octads, respectively. It is clear that -W,, is precisely the subgroup of the symmetric group on Q which preserves V~ * This

result

has

also

been

obtained

by

D.

173 Copyright .W rights

.C 1974 by Academic Press, Inc. of reproduction in any form reserx-ed.

Wright

2nd

T.

Yoshida.

174

LARRY

FINKELSTEIN

Let Rz4 be spanned by the orthonormal basis u’i (zIE Q) and define 71~as x zji (i E S) whenever S C Q. The Leech lattice fl is spanned by the vectors vc (C E Vs) together with the vector vo - 4v, . The group .O is defined to be the group of all isometries of Rz4 which preserve /l as a whole. . 0 is a perfect group of order 2”’ . 3g . 5a . 72 . 1 I . 13 . 23 and has center {&I}. The central quotient group of .O is the simple group . 1. An important subgroup of . 0 is the maximal 2-local subgroup N which we describe as follows. A permutation Z- of 8 may be extended to an isometry of R24 by defining vin = vi,, and if S _C9 we defme the isometry e, of R2” by vies = vi (i $ S) or -vi (i E S). TABLE

. 0 Class

Centralizer

I

in . 0

Squares

2-J 2B 2c 20 44

2B

4B

20

4C

2B

40

2c

4E

20

4F

2A

4G

20

4H

2B

8A

4c

8B

4G

8C

4c

80

4E

8E

4H

8F

4A

8G

4F

8H

4G

81

4c

16A

8C

16B

81

16C

8C

FINITE

145

GROUPS

Observe that g and e, belong to . 0 precisely wher, “r c .I<;, and S E ‘g: respectively. We define N to be the subgroup of . 0 generated by these elements. The subgroup E generated by the elements e, (S E %) is a normai elementary abelian subgroup of order 2’2 cf N and has a complement isomorphic 10 M2, . Th e orbits of N acting by cosjcgation on I? = J? - ,[ ij are the involutions eD , e, (S E g8), e, (S E ‘t;,,), a.r,d es (S E %I&. The character table of 0 has been constructed by T. Conway, XI. Cw-, X. Patterson, and J. Thompson (unpublished). IJTe shali need to know the classes of elements of 2-power order of . 0 and list this information in Table I. if E E e8 and C E %,, ) then eR , eB , e, , and eBPD betong to the classes 2-4, 2B, 2C, and 2D, respectively. The structure of the centraiizers of elements of 0 wili be described as needed.

PDX$ ?iTe may choose T = e, (C E 44r,,) and observe that §s?ppose E1 E gIt(. 0), and 7 E E1 . Since WI~(M& = 3, m(Et’E n El) = nz(EE,jE) .<. 3 and so 1E n El 1 > 2”. But of Mx,, fixes subspaces of E of order 28 or 26 accordingly structure 1828 or 2l* on Q. This implies that El = E and the LmI3i.x

1.2.

Let S ~5Svl,(. - - O), ti fez Z(S)

C+(T) = Eli?:, _ it focllows that an invoiution 77 as 7~ has cycie ~eesuit is proved.

is eleme-ntay: abeiian sf order 4.

PIXW~-. -%ssuming S < A’* whence E < S, it foiiows f:om the acrion oi A”sfz4on % that a S+w 2 subgroup of Mfi,, fixes precisely one elemen: cf g* and its complement in 9Y16and none of @,, . Thus Z(S) mtist be as described. The following result is a direct cow,equence wiil be used in conjunction with Lemma I. I.

Pqf~

of -4.iperir;‘s theorem

See Lemma 3.2 of [il.

2. PROOF OF THECXEXE

The proof is divided

into several parts. Recall that I-I = C,(a) 2

0

and

176

LARRY

FINKELSTEIN

Proof. Assume the contrary and suppose that for some T E Syl,(H) and SE Syl,(G), T < S. By Lemma 1.2, 3 = (2, t) = Z(T) is elementary abelian of order 4. Moreover, T < S implies that T < A&(T) so that one can find an element g E N,(T) such that .sg = t (or possibly zt). Since elements of 4F square to z, there exists in T a square root w of a, whence (ZUQ)* = (zua)g = .sg = t. Therefore, CJ(W, tj)g = Cc((w, tj)g = Cc() and C,(zug) have different orders. subgroup Now C,(W) has order 215 . 33 . 5’ . 7 . 13 and contains a Fal K of index 2 isomorphic to the central product of Z, by G,(4), the proper 2 fold cover of G,(4), with an element of C,(W) 2 effecting the field automorphism on the central quotient group of G,(4). Since C,(w”) is conjugate to a subgroup of C,(w) and since (zuQ)* = t lies in 2B or 20, it follows from an examination of Table 1 that wg lies in one of the classes 4B, 4E or 4G. Hence ) C&w”)\ = 2r5 . 3a . 5 or 215 . 3”. To complete the proof it suffices to show that 3” ) 1 CH((;w, t:>)l, i.e, that t does not centralize a subgroup U of order 9 of CH(w). Such a subgroup U must lie in K. It is easily established (cf. [3]) that a subgroup of order 9 of G,(4) is self centralizing. Hence by the known structure of K we have C,(U) = Z, x li. Moreover, the inequality [CH((w, Uj): C,(U)] ,( 2 together with the fact, from Table I, that some element of 8G centralizes a Sylow 3 subgroup of CH(zu) gives CH((w, U>) = Za x U. This excludes the possibility of t centralizing U and supplies us with the necessary contradiction in order to complete the proof. From nom on assume that SE SyI,(H). At this stage we can begin our analysis of the fusion of s in G to an element of H knowing from Lemma 2.1 that S E Syl,(G) as well. Lmmu

2.2.

z canaot fuse in G to an element of 2B or 20.

Proof. Suppose t (a, t> = Z(S). Since g E IV,(S) which acts and must, according eighth root in S. This LEivmA

2.3.

lies in 2B or 20 and fuses in G to .s. We may assume AT,(S) controls fusion in Z(S) there exists an element cyclically on the set (2, zt, t}, But either t or tx lies in 2B to Table I have an eighth root in S whereas z has no is a contradiction and the result follows.

z cannot fuse in G to an element of 2C.

Proof. It is clear from Lemma 2.2 that if z fuses in G to a noncentral element ,8 of S, then /3 must lie in 2C. Choosing /J $ Z(S) fused in G to z such that / C,(p)1 is maximal, it follows from Lemma 1.3 and the fact that x is isolated in Z(S) that there is an element p’ $Z(S), a Sylow 2 intersection

FINITE

GROUPS

1-p

S n T = C&Y) conjugate in G to C,(p) and a 2 elemen: x E A’G(S n T) such that/Y” = .z. Sisce 1 C,(/3’)1 is maximal subject to ,6’ $ Z(S), p’ fused ir_ G r~ u?, C&2’) must be a Sylow 2 subgroup of C,(f’). R’iloreorer, according to Lemma 1‘ 1, C,(p’) contains precisely one subgroup E’ E 6,,[N) so that z and /3’ must be conjugate in Ai, by a 2 elemenr. Now E’ is conjugate in H TOE = O,(;V), so we may assume that /3’ = ec (C E el,), x = ps and iJ’ is conjugate to 2 in NG(E). From our description of the orbits of :V acting on E- ir follows That :V6{E) acting on E” must have an orbit A of length 2577 consisting of the element P~ together v&h the 2576 elements e, (C E ??& But 2577 has prime facto&ation 3.859 whence 8.59 1 1lVG(E)/C,(E)~, contrary to rhe fact that 859 1 1 5X&2)1. This contradiction establishes our resu!t. -1ccording to Table I and the results of Lemma 2.2 al;id 2.3, it folio~~~s that z is isokted in N. Therefore, by Glailberman’s Z” theorem, zO(G) is central in G/Q(G), and we conclude that G = O(G)H.

The author would like to thank in tile proof of Theorem 2.1.

R. Solomon

for several suggestions

whicL1 helped

REFERENCES I. P. CH.EiOT, Groups whose Sylow 2 subgroups ha\.-e cyclic commutator groups, J. .Qebun I9 (1971), 21-30. 2. J. H. CONWAY, A group of order 8, 315, 553, 613, 086, 720, EM. Lo&~~r X&h. SOC. 1 (1969), 79-88. 3. H. Er;onioTo, The conjugacy classes of ChevalIeJ- groups of type (G,) over finite fields of characreristic 2 or 3, J. Fnc. Sci. Uniz. Tok>v 16 (I?70), 497-512. 4, G. GLXUBERMXN, Central eiements in core-free groups, J. .$&bra 4 j1966), 403-610. 5. D. GOKEKSTEIN, “Finite Groups,” Harper and Rowl New York, 1968.