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

Abstract

Abstract For any given subgroup H of a finite group G, the Quillen poset ${\mathcal {A}}_p(G)$ of nontrivial elementary abelian p-subgroups is obtained from ${\mathcal {A}}_p(H)$ by attaching elements via their centralisers in H. We exploit this idea to study Quillen’s conjecture, which asserts that if ${\mathcal {A}}_p(G)$ is contractible then G has a nontrivial normal p-subgroup. We prove that the original conjecture is equivalent to the ${{\mathbb {Z}}}$ -acyclic version of the conjecture (obtained by replacing ‘contractible’ by ‘ ${{\mathbb {Z}}}$ -acyclic’). We also work with the ${\mathbb {Q}}$ -acyclic (strong) version of the conjecture, reducing its study to extensions of direct products of simple groups of p-rank at least $2$ . This allows us to extend results of Aschbacher and Smith and to establish the strong conjecture for groups of p-rank at most $4$ .