The homotopy types of the posets of p-subgroups of a finite group

2019

Journal of Algebra

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We investigate a stronger formulation of Webb's conjecture on the contractibilty of the orbit space of the p-subgroup complexes in terms of finite topological spaces. The original conjecture, which was first proved by Symonds and, more recently, by Bux, Libman and Linckelmann, can be restated in terms of the topology of certain finite spaces. We propose a stronger conjecture, and prove various particular cases by combining fusion theory of finite groups and homotopy theory of finite spaces.

Section: Contractibility Of a P (G) ′ /Gmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A stronger reformulation of Webb's conjecture in terms of finite topological spaces

2019

Journal of Algebra

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“…Also, the homology groups and the fundamental group of the posets are those of their associated complexes. Note that this is not the convention that we adopted in our previous articles [23, 24]. In those papers, we handled the posets of

p

‐subgroups as finite topological spaces, with an intrinsic topology, where the notion of homotopy equivalence is strictly stronger than in the context of simplicial complexes.…”

Section: Introductionmentioning

confidence: 99%

The fundamental group of the p‐subgroup complex

Minian

2020

J. London Math. Soc.

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We study the fundamental group of the p‐subgroup complex of a finite group G. We show first that π1false(A3(A10)false) is not a free group (here A10 is the alternating group on ten letters). This is the first concrete example in the literature of a p‐subgroup complex with non‐free fundamental group. We prove that, modulo a well‐known conjecture of Aschbacher, π1false(Ap(G)false)=π1false(Ap(SG)false)∗F, where F is a free group and π1false(Ap(SG)false) is free if SG is not almost simple. Here SG=normalΩ1false(Gfalse)/Op′false(Ω1(G)false). This result essentially reduces the study of the fundamental group of p‐subgroup complexes to the almost simple case. We also exhibit various families of almost simple groups whose p‐subgroup complexes have free fundamental group.

“…For example, condition 1 is a generalisation to every prime of the analogous result [5,Proposition 1.6] stated for > 5, and in our proof we use the classification of simple groups to a much lesser extent. Condition 2 of the theorem is based on our previous work on the fundamental group [11]. The more technical hypotheses of conditions 3 and 4 are focussed on extending [5,Main Theorem] to every odd prime .…”

mentioning

confidence: 99%

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

Forum of Mathematics, Sigma

2021

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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 $4$ .