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On the maximal overgroups of Sylow subgroups of finite groups

Barbara Baumeister, Timothy C. Burness, Robert M. Guralnick, Hung P. Tong-Viet

Abstract

In this paper, we determine the finite groups with a Sylow $r$-subgroup contained in a unique maximal subgroup. The proof involves a reduction to almost simple groups, and our main theorem extends earlier work of Aschbacher in the special case $r=2$. Several applications are presented. This includes some new results on weakly subnormal subgroups of finite groups, which can be used to study variations of the Baer-Suzuki theorem.

On the maximal overgroups of Sylow subgroups of finite groups

Abstract

In this paper, we determine the finite groups with a Sylow -subgroup contained in a unique maximal subgroup. The proof involves a reduction to almost simple groups, and our main theorem extends earlier work of Aschbacher in the special case . Several applications are presented. This includes some new results on weakly subnormal subgroups of finite groups, which can be used to study variations of the Baer-Suzuki theorem.
Paper Structure (17 sections, 54 theorems, 50 equations, 9 tables)

This paper contains 17 sections, 54 theorems, 50 equations, 9 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Prime divisors
  4. Almost simple groups
  5. Subgroup structure
  6. The reduction theorem
  7. Sporadic and alternating groups
  8. Proof of Theorem \ref{['t:prime2']}
  9. Proof of Theorem \ref{['t:main1']}: Classical groups
  10. Linear groups
  11. Unitary groups
  12. Symplectic groups
  13. Odd dimensional orthogonal groups
  14. Even dimensional orthogonal groups
  15. Proof of Theorem \ref{['t:main1']}: Exceptional groups
  16. ...and 2 more sections

Key Result

Theorem 1

Let $G$ be an almost simple group with socle $T$, let $R$ be a Sylow $2$-subgroup of $G$ and assume $G/T$ is a $2$-group. Then $\mathcal{M}(R) = \{H\}$ if and only if one of the following holds:

Theorems & Definitions (107)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Corollary 5
  • Corollary 6
  • Theorem 7
  • Corollary 8
  • Lemma 2.1
  • proof
  • ...and 97 more