On the maximal overgroups of Sylow subgroups of finite groups

TL;DR

This paper classifies finite groups where a Sylow subgroup is contained in a unique maximal subgroup, extending previous work and providing applications to subgroup structure and the Baer-Suzuki theorem.

Contribution

It extends Aschbacher's results to all primes r, offering a comprehensive classification of such groups and new insights into subgroup properties.

Findings

Classification of groups with unique maximal Sylow overgroups

Extension of Aschbacher's theorem to all primes r

New results on weakly subnormal subgroups

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.