Some results on Quillen’s conjecture via equivalent-poset techniques

Abstract

We extend the main theorem of Aschbacher and Smith on the Quillen conjecture from $p>5$ to the remaining odd primes $p=3,5$. In the process, we develop further combinatorial and homotopical methods for studying the poset of non-trivial elementary abelian $p$-subgroups of a finite group. The techniques lead to a number of further results on the conjecture, often reducing dependence on the CFSG; in particular, we also provide some partial results toward the case of $p=2$.