Abstract

Let Ap(G) be the poset of non-trivial elementary abelian p-subgroups of a finite group G at a given prime p. Daniel Quillen established important connections between intrinsic algebraic properties of G and homotopical properties of Ap(G). For example, he showed that Ap(G) is disconnected if and only if G contains a strongly p-embedded subgroup, and that Ap(G) is contractible if G contains a non-trivial normal p-subgroup. He conjectured the converse of the latter giving rise to the well-known Quillen’s conjecture. Although there has been significant progress on the conjecture, it is still open. One of the major advances was achieved by Michael Aschbacher and Stephen D. Smith: they proved that the conjecture holds for p>5, under certain restrictions on finite unitary groups.In this talk, we will see some techniques to understand the homotopy type of the poset Ap(G) from a subposet Ap(H), where H is some subgroup of G. This will allow us to perform homotopical-replacements of Ap(G) by non-standard p-subgroup posets, which leads to new ways of understanding the homotopy type of the Ap-posets. As a consequence of these methods, I will present new developments on Quillen’s conjecture: the extension of Aschbacher-Smith’s theorem to every odd prime p, and also to p=2 (under certain restrictions on some families of simple groups of Lie type). These results were obtained in collaboration with Stephen D. Smith.