Large maximal subgroups of finite almost simple groups

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 maximal subgroups whose index coincides with the smallest index $\operatorname{l}(S)$ of a maximal group of $S$ or there exists a natural number $v_S \leq {\operatorname{l}(S)^2}$, depending only on $S$, such that $R$ has a conjugacy class of core-free maximal subgroups with index $v_S$. We show that the number of subgroups of the outer automorphism group of $S$ is bounded by $\log^3 {\operatorname{l}(S)}$ and $\operatorname{l}(S)^2 < |S|$.