An approach to Quillen's conjecture via centralisers of simple groups

Abstract

For any given subgroup H of a finite group G, the Quillen poset of nontrivial elementary abelian p-subgroups is obtained from by attaching elements via their centralisers in H. We exploit this idea to study Quillen's conjecture, which asserts that if is contractible then G has a nontrivial normal p-subgroup. We prove that the original conjecture is equivalent to the-Acyclic version of the conjecture (obtained by replacing 'contractible' by '-Acyclic'). We also work with the-Acyclic (strong) version of the conjecture, reducing its study to extensions of direct products of simple groups of p-rank at least. This allows us to extend results of Aschbacher and Smith and to establish the strong conjecture for groups of p-rank at most.