The Radical 2-Subgroups of the Sporadic Simple Groups J4, Co2, and Th

Abstract

For a finite group G and a prime divisor p of its order, a nontrivial Ž Ž .. p-subgroup R of G is called radical, if O N R R. With the poset p G Ž . B G of radical p-subgroups of G with respect to inclusion we naturally p Ž Ž .. associate the simplicial complex of its chains, denoted B G , which is p known to be G-homotopy equivalent to the complex associated with the poset of nontrivial p-subgroups as well as that of nontrivial elementary Ž . abelian p-subgroups of G see, e.g., 5, 6.6 . The latter are important tools Ž to investigate the structure of G concerning the prime p see, e.g., 5, . Ž Ž .. Chap. 6 . Thus the smaller complex B G has enough information to p understand behaviors of G on p. The following fact, a corollary of the Borel Tits theorem 5, 6.8.4 , Ž Ž .. shows the further importance of B G : for a finite group of Lie type p Ž Ž .. defined over a field in characteristic p, the complex B G is the p barycentric subdivision of the building for G. Thus for an arbitrary finite Ž Ž .. group G we may think of the complex B G as a generalization of the p concept of buildings: in fact, as is observed in 14 , for each sporadic simple Ž Ž .. group G of characteristic-p type, B G is homotopy equivalent to p 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 equivaŽ Ž .. Ž . Ž lent to B G for every specifically sporadic simple group G see also p . the introduction of 17 .