Maximal subgroups of small index of finite almost simple groups

TL;DR

This paper investigates the structure of maximal subgroups of finite almost simple groups, establishing bounds on their indices and the number of certain subgroups, which advances understanding of their subgroup composition.

Contribution

It proves the existence of specific conjugacy classes of core-free maximal subgroups with bounded indices in finite almost simple groups, depending only on the socle.

Findings

Existence of conjugacy classes of core-free maximal subgroups with index equal to the minimal index of maximal subgroups of S.

Bound on the number of subgroups of the outer automorphism group of S, specifically by log^3 l(S).

Inequality l(S)^2 < |S|, relating the minimal index to the size of S.

Abstract

We prove in this paper that a finite almost simple group $R$ with socle the non-abelian simple group $S$ possesses a conjugacy class of core-free maximal subgroups whose index coincides with the smallest index $\operatorname{l}(S)$ of a maximal group of $S$ or a conjugacy class of core-free maximal subgroups with a fixed index $v_S \leq {\operatorname{l}(S)^2}$, depending only on $S$. We show that the number of subgroups of the outer automorphism group of $S$ is bounded by $\log^3 {\operatorname{l}(S)}$ and $\operatorname{l}(S)^2 < |S|$.