Eliminating components in Quillen's conjecture

Piterman, Kevin Iván; Smith, Stephen D.

doi:10.1016/j.jalgebra.2021.05.011

Cited by 2 publications

(2 citation statements)

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“…Finally, these computations show that A 2 ( ) ↩→ A 2 (Aut( )) induces an inclusion in homology, and hence a non-zero map. By the main result of [17], PSL 2 (9) is not a component of a minimal counterexample to Quillen's conjecture.…”

Section: Group Number Of Involutions Number Of 4-subgroupsmentioning

confidence: 99%

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Maximal subgroups of exceptional groups and Quillen's dimension

Piterman¹

2023

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A. Given a finite group and a prime , let A ( ) be the poset of nontrivial elementary abelian -subgroups of . The group satisfies the Quillen dimension property at if A ( ) has non-zero homology in the maximal possible degree, which is the -rank of minus 1. For example, D. Quillen showed that solvable groups with trivial -core satisfy this property, and later, M. Aschbacher and S.D. Smith provided a list of all -extensions of simple groups that may fail this property if is odd. In particular, a group with this property satisfies Quillen's conjecture:has trivial -core and the poset A ( ) is not contractible.In this article, we focus on the prime = 2 and prove that the 2-extensions of the exceptional finite simple groups of 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 = 2.

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Section: Group Number Of Involutions Number Of 4-subgroupsmentioning

confidence: 99%

“…In the last few years, there have been further developments in the Quillen conjecture [15,16,17,18]. Recently, in [18], new tools for the study of the conjecture have been provided.…”

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confidence: 99%