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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

Given a finite group G and a prime p, let 𝒜p(G) be the poset of nontrivial elementary abelian p-subgroups of G. The group G satisfies the Quillen dimension property at p if 𝒜p(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 𝒜p(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.

$p$-subgroups, exceptional groups of Lie type, Quillen's conjecture
Mathematical Subject Classification
Primary: 05E18, 20D20, 20D30, 20G41
Received: 16 January 2023
Revised: 19 May 2023
Accepted: 3 September 2023
Published: 13 June 2024
Kevin I. Piterman
Fachbereich Mathematik und Informatik
Philipps-Universität Marburg

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