Generating maximal subgroups of finite almost simple groups

TL;DR

This paper establishes tight bounds on the minimal number of generators for maximal subgroups of finite almost simple groups, improving previous results by using the theory of crowns in finite groups.

Contribution

It provides a precise upper bound of 5 and characterizes when the bound is at least 4 for maximal subgroups of finite almost simple groups.

Findings

d(H)  5 for maximal subgroups H

d(H)  4 if and only if H is in a known list

Improves previous bounds by Burness, Liebeck, and Shalev

Abstract

For a finite group $G$, let $d(G)$ denote the minimal number of elements required to generate $G$. In this paper, given a finite almost simple group $G$ and any maximal subgroup $H$ of $G$, we determine a precise upper bound for $d(H)$. In particular, we show that $d(H)\leq 5$, and that $d(H)\geq 4$ if and only if $H$ occurs in a known list. This improves a result of Burness, Liebeck and Shalev. The method involves the theory of crowns in finite groups.