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

Part of: Algebraic combinatorics Ordered sets Connections with homological algebra and category theory Representation theory of groups

Published online by Cambridge University Press:  07 June 2021

Kevin Iván Piterman Open the ORCID record for Kevin Iván Piterman [Opens in a new window]
Kevin Iván Piterman*
Affiliation:
Departamento de Matemática, IMAS-CONICET, FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina; E-mail: kpiterman@dm.uba.ar.

Abstract

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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 $4$.

MSC classification

Primary: 06A11: Algebraic aspects of posets 20D30: Series and lattices of subgroups 20J05: Homological methods in group theory
Secondary: 05E18: Group actions on combinatorial structures 20D05: Finite simple groups and their classification 20D25: Special subgroups (Frattini, Fitting, etc.)
Type
Algebra
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e48
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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