The fundamental group of the p‐subgroup complex

Minian, Elías Gabriel; Piterman, Kevin Iván

doi:10.1112/jlms.12380

Cited by 2 publications

(2 citation statements)

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“…We give some applications of Theorem 4.1. The following results depend on the results obtained on the fundamental group of the -subgroup posets [12] and the almost-simple case of the conjecture [4]. Proof.…”

Section: Proof Ifmentioning

confidence: 94%

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

Piterman

2021

Forum of Mathematics, Sigma

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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 $4$ .

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Section: Proof Ifmentioning

confidence: 94%

“…In this way, we can work with smaller subposets and apply inductive arguments. This approach has its roots in our previous work with Minian on the fundamental group of these complexes [12]. The poset N( ) was also considered by Segev and Webb [18,19].…”

Section: Introductionmentioning

confidence: 98%