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The Radical 2-Subgroups of the Sporadic Simple Groups J (4), Co2, and Th

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The Radical 2-Subgroups of the Sporadic Simple Groups J (4), Co2, and Th Journal of Algebra 233, 309᎐341Ž. 2000 doi:10.1006rjabr.2000.8429, available online at http:rrwww.idealibrary.com on The Radical 2-Subgroups of the Sporadic Simple Groups J4 , Co2, and Th Satoshi Yoshiara CORE Metadata, citation and similar papers at core.ac.uk Di¨ision of Mathematical Sciences, Osaka Kyoiku Uni¨ersity, Kashiwara, Provided by Elsevier - Publisher Connector Osaka 582-8582, Japan E-mail: [email protected] Communicated by Gernot Stroth Received December 13, 1999 1. INTRODUCTION For a finite group G and a prime divisor p of its order, a nontrivial s p-subgroup R of G is called radical,ifONpGŽŽ.. R R. With the poset BpŽ.G of radical p-subgroups of G with respect to inclusion we naturally ⌬ associate the simplicial complex of its chains, denoted ŽŽ..Bp G , which is known to be G-homotopy equivalent to the complex associated with the poset of nontrivial p-subgroups as well as that of nontrivial elementary abelian p-subgroups of G Žsee, e.g.,wx 5, 6.6. The latter are important tools to investigate the structure of G concerning the prime p Žsee, e.g.,w 5, x ⌬ Chap. 6.Ž . Thus the smaller complex BpŽG..has enough information to understand behaviors of G on p. The following fact, a corollary of the Borel᎐Tits theoremwx 5, 6.8.4 , ⌬ shows the further importance of ŽŽ..Bp G : for a finite group of Lie type ⌬ defined over a field in characteristic p, the complex ŽŽ..Bp G is the barycentric subdivision of the building for G. Thus for an arbitrary finite ⌬ group G we may think of the complex ŽŽ..Bp G as a generalization of the concept of buildings: in fact, as is observed inwx 14 , for each sporadic simple ⌬ group G of characteristic-p type, ŽŽ..Bp G is homotopy equivalent to some smaller simplicial complex, previously constructed by an ad hoc method and referred to as a ‘‘p-local geometry’’ of G. This suggests the importance of examining minimal complexes which are homotopy equiva- ⌬ lent to ŽBp ŽG ..for every Ž specifically sporadic simple.Ž group G see also the introduction ofwx 17. 309 0021-8693r00 $35.00 Copyright ᮊ 2000 by Academic Press All rights of reproduction in any form reserved. 310 SATOSHI YOSHIARA To investigate such complexes, we first need to determine BpŽ.G for every sporadic simple group G and a prime divisor p of <<G . For small sporadics, such works appeared inwx 1Ž. for the five Mathieu groups ,wx 8Ž for wx wx J3., 2Ž. for He , and 3Ž. for Co2 during the verification of the Dade s conjecture. The posets B2Ž.Rud and Bp Ž.Suz for p 2 and 3 are deter- mined inwx 13, 17 , respectively. For a fairly large sporadic group Co1 and s wx p 2 and 3, Sawabe classified BpŽ.Co1 12, 13 . The author was also recently informed from K. Uno that the Dade conjecture is verified for J2 , McL, and HS. This paper is the first one in a series of papers determining BpŽ.G for every sporadic simple group G and a prime divisor p of <<G , which are not mentioned above. Here the classification of B2Ž.G up to conjugacy is given for every sporadic simple group of characteristic-2 type, except those already treated by the other authors. Among 10 sporadics of characteristic-2 type, B21Ž.J is easy to determine Ž it has two classes with representatives a subgroup of order 2 and 8. , because a Sylow 2-subgroup of J1 is elemen- 3 tary abelian of order 8 with normalizer E87: F , which is maximal in J1: the wx five Mathieu groups and J3 are treated in 1, 8 , respectivelyŽ for odd s primes p as well.Ž . Thus in this paper, we classify B24G.for G J , Co2, and Th. The author includes B2Ž.Co2 , though it has already been deter- mined inwx 3 , because the methods in the present paper give more geomet- ric and explicit descriptions. Sawabe and S. D. Smith also obtained the same results as well as the classification of radical 2-chains for J4 and Co2. The information in the present paper and inwx 10, 18, 19 on radical subgroups for sporadic simple groups is used by Sawabe for determining smaller subclasses of radical chains to which the verification of the Dade conjecture can be reduced. His results are based on interesting general observations, which can be regarded as a generalization of some fundamental lemmas in Section 2 to radical chains. wx In the subsequent papers 10, 18, 19 , we classify BpŽ.G for the Fischer X s s X groups Fi22, Fi 23, Fi 24 and p 2, 3; for G Ly, O N, HS, and HN and p s 2Ž here the author includes HS, as the result is required for classifying B2Ž..HN ; and for all sporadic simple groups G and odd primes p, respectively. Thus at the time of writing, the only remaining cases are the Monster M and the Baby Monster BM for p s 2. The principle of the classification is the inductive method proposed in wx14 and discussed in wx 16Ž see alsowx 12. , which is efficient if we know the list of maximal p-local subgroups of G as well as inclusions of some p-local subgroups up to conjugacy. Our method is purely group theoretic and computer-free, while the method presented inwx 3 requires a permuta- RADICAL 2-SUBGROUPS 311 tion representation of the quotient group of each maximal p-local sub- group by its Op-part to examine all p-subgroups of that quotient group. In Section 2, we prepare some basic technical tools which are frequently used inwx 10, 18, 19 as well. Some of them already appeared in wx 12 in a slightly different form. The most convenient one is Lemma 1Ž. 4 saying that r the inverse images of radical subgroups of M OMpŽ.are radical if a maximal p-local subgroup M has extraspecial OMpŽ.. The geometric descriptions of the radical 2-subgroups of the Mathieu groups M24 and wx M22 Žwith the correction of the results in 16. are included in this section as well as that of their actions on the Golay code and cocode. They are of fundamental importance in Sections 3, 4 and the paperwx 10 . In Sections ᎐ 3 5, the groups J4 , Co2, and Th are treated independently. Each of these sections gives brief descriptions of maximal 2-local subgroups of the simple group in question. 2. PRELIMINARIES 2.1. Some Lemmas In this subsection, the author collects some lemmas which will be frequently used later. LEMMA 1. Let G be a finite group and p a prime di¨isor of<< G . Ž.1wx 14, Lemma 1.9 If a radical p-group R of G is contained in a F r subgroup M of G, then OppŽ. M R and R OŽ. M is a radical p-subgroup of r ¨ M OpŽ. M or the tri ial group. Ž.2 ŽA ¨ariant of wx12, Lemma 2.1 . Assume that a p-subgroup R contains OpŽ. M as a normal subgroup for some subgroup M of G and that r r F R OppŽ. M is a radical p-subgroup of M OMŽ.. If NG Ž. R M, then R is a radical p-subgroup of G. Ž.3 Let Z be a nontri¨ial p-subgroup of G and M be a maximal p-local [ r subgroup of G containing NGŽ. Z . Set R0 OpipŽ. M and let R OMŽ. s ¨ r Ž.i 1,...,m be the complete representati es of M OMpŽ.-classes of radical r [ ¬ s s p-subgroups of M OMpjjŽ.. Then the set S Ä R j 0,...,m, ZRŽ. Z4 is a system of complete representati¨es of radical p-subgroups R whose centers ZŽ. R are conjugate to Z. s Ž4 .Assume that OpG Ž C Ž z .. is non-abelian and ² z :ZO ŽpG Ž C Ž z ... ¨ ¨ for a p-element z. Then the in erse images in CGŽ. z of the representati es of r classes of radical p-subgroups of CGpGŽ.ŽŽ.. z O C z together with OpGŽŽ.. C z form a set of complete representati¨es of radical p-subgroups R with ZŽ. R generated by a conjugate of z. 312 SATOSHI YOSHIARA r Proof. Ž.2 The group R OMp Ž .is trivial or a radical p-subgroup of r s F M OMppŽ.exactly when ONŽMG Ž.. R R.As NRŽ. M, we have s s g ONpGŽ Ž R ..ON pM Ž Ž R .. R, which implies that R BpŽ.G . F Ž.3 Each member R of S has the center Z, so that NRGŽ. s F NZRGGŽŽ..NZ Ž. M. Then it follows from the claimŽ. 2 that R is a radical p-subgroup of G with center Z. Conversely, let R be a radical g s g g F p-subgroup of G with ZRŽ. Z for some g G. Then NRGŽ . g s F 1 g NZRGGŽŽ ..NZ Ž . M. It follows fromŽ.Ž. 1 that OMpy R and either g s gr r R OMppŽ.or R OMŽ.is a radical p-subgroup of M OMpŽ.. Then R g is conjugate to exactly one member of S under M. Distinct members Rijand R of S are not conjugate under G, because s h g s s if RijR for some h G then h normalizes ZRŽ.iZR Ž.
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