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On the homotopy type of p-subgroup posets

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GR2W02 - Simple groups, representations and applications

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.

This talk is part of the Isaac Newton Institute Seminar Series series.

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