On the maximal subgroups of almost simple and primitive perfect groups

TL;DR

The paper proves a uniform bound on the derived length of maximal subgroups of finite almost simple groups and extends to primitive perfect groups: for any almost simple $G$ with maximal subgroup $H$, the $10$th derived term $H^{(10)}$ is perfect, with $H^{(9)}$ also perfect except when $H$ is a solvable maximal subgroup of Fi$_{23}$. The argument hinges on a generalized derived length $\mathrm{dl^*}$ and a key lemma: if $N\unlhd H$ and $N^{(a)}$ and $(H/N)^{(b)}$ are perfect, then $H^{(a+b)}$ is perfect, which guides a case analysis across socles $T$ (alternating, classical, exceptional, sporadic) and parabolic subgroups, using CFSG toolkits (Aschbacher’s classification, O'Nan–Scott) and Atlas data for sporadics. The results are sharp for the maximal subgroups of Fi$_{23}$ and reveal obstructions when extending to primitive perfect groups; nevertheless, the paper shows that, in general, a universal constant bound governs the derived length escalation of stabilizers in primitive actions. Overall, the work provides a cohesive, CFSG-based framework for understanding the structural depth of maximal subgroups in a broad class of finite 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.

Theorems & Definitions (35)

• Theorem 1.1
• Example 1.2
• Theorem 1.3
• Conjecture 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3
• proof