On the maximal subgroups of almost simple and primitive perfect groups

TL;DR

This paper proves that for finite almost simple groups and their maximal subgroups, the 10th derived subgroup is perfect, establishing a precise bound that is proven to be optimal.

Contribution

It establishes a sharp bound on the derived series of maximal subgroups in almost simple and perfect groups, advancing understanding of their subgroup structure.

Findings

The 10th derived subgroup of maximal subgroups in almost simple groups is perfect.

The same property holds for core-free subgroups in perfect groups.

The bound of 10 is proven to be the best possible.

Abstract

We prove that, if $G$ is a finite almost simple group and $H$ is a maximal subgroup of $G$, then the $10$th term of the derived series of $H$ is perfect. The same is true if $G$ is perfect and $H$ is core-free. The constant $10$ is best possible.