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

Abstract

We show that, 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 centralizers in $H$. We use 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 order divisible by $p$ and $p$-rank at least $2$. This allows to extend results of Aschbacher-Smith and to establish the strong conjecture for groups of $p$-rank at most $4$.