Maximal subgroups of exceptional groups and Quillen’s dimension

Piterman, Kevin I.

doi:10.2140/ant.2024.18.1375

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Maximal subgroups of exceptional groups and Quillen's dimension

Kevin I. Piterman

Vol. 18 (2024), No. 7, 1375–1401

DOI: 10.2140/ant.2024.18.1375

Abstract

Given a finite group $G$ and a prime $p$ , let $𝒜_{p} (G)$ be the poset of nontrivial elementary abelian $p$ -subgroups of $G$ . The group $G$ satisfies the Quillen dimension property at $p$ if $𝒜_{p} (G)$ has nonzero homology in the maximal possible degree, which is the $p$ -rank of $G$ minus $1$ . For example, D. Quillen showed that solvable groups with trivial $p$ -core satisfy this property, and later, M. Aschbacher and S. D. Smith provided a list of all $p$ -extensions of simple groups that may fail this property if $p$ is odd. In particular, a group $G$ with this property satisfies Quillen’s conjecture: $G$ has trivial $p$ -core and the poset $𝒜_{p} (G)$ is not contractible.

In this article, we focus on the prime $p = 2$ and prove that the $2$ -extensions of finite simple groups of exceptional Lie type in odd characteristic satisfy the Quillen dimension property, with only finitely many exceptions. We achieve these conclusions by studying maximal subgroups and usually reducing the problem to the same question in small linear groups, where we establish this property via counting arguments. As a corollary, we reduce the list of possible components in a minimal counterexample to Quillen’s conjecture at $p = 2$ .

Keywords

$p$-subgroups, exceptional groups of Lie type, Quillen's conjecture

Mathematical Subject Classification

Primary: 05E18, 20D20, 20D30, 20G41

Milestones

Received: 16 January 2023

Revised: 19 May 2023

Accepted: 3 September 2023

Published: 13 June 2024

Authors

Kevin I. Piterman
Fachbereich Mathematik und Informatik Philipps-Universität Marburg Marburg Germany

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Vol. 18, No. 7, 2024