On the maximal subgroups of almost simple and primitive perfect groups

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.

18 pages, presentation improved, to appear in Math. Proc. Cambridge Philos. Soc