New Collections ofp-Subgroups and Homology Decompositions for Classifying Spaces of Finite Groups

Abstract

We define new collections of p-subgroups for a finite group G and p a prime dividing its order. We study the homotopy relations among them and with the standard collections of p-subgroups and determine their ampleness and sharpness properties.

On fixed point sets and Lefschetz modules for sporadic simple groups

“…We will refer to a p-group as being distinguished if it contains a p-central element in its center; see the paper [21] for further information on distinguished collections. For odd primes, we analyze in detail the distinguished p-radical complexes for those sporadic simple groups which have a Sylow p-subgroup of order p 3 .…”

“…The groups McL (with p = 5) and O N (with p = 7) have local characteristic p; all the p-local subgroups H satisfy the condition C H (O p (H)) ≤ O p (H).In all these cases, the distinguished Bouc collection equals the whole Bouc collection and therefore the reduced Lefschetz modules are projective. For a proof of the equality of the two collections in a group of local characteristic p see[21, Lemma 4.8]. Information on the p-radical subgroups for odd primes, for G one of the sporadic simple groups, is given in[41, …”

On fixed point sets of distinguished collections for groups of parabolic characteristic

“…This paper continues the systematic study, started in [11], of certain collections of p-subgroups, which we call distinguished. These are subcollections of the standard collections of p-subgroups and which consist of those p-subgroups which contain p-central elements in their centers.…”

On a homotopy equivalence between the 2-local geometry and the Bouc complex for the sporadic group McL

“…We prove that a certain subcomplex B I 2 of the Bouc complex, which is isomorphic to a subdivision of the geometry ∆, is homotopy equivalent to the full Bouc complex B 2 (in fact, this is a retract under an equivariant deformation retraction). Our approach is similar to the one used in our previous work [6], which describes such a relationship for the sporadic group Co 3 , although for Co 3 the standard 2-local geometry is homotopy equivalent to the "distinguished" Bouc collection; see [7].…”