Maximal subgroups of small index of finite almost simple groups

Abstract

We prove in this paper that every almost simple group R with socle isomorphic to a simple group S possesses a conjugacy class of core-free maximal subgroups whose index coincides with the smallest index {{,mathrm{l},}}(S) of a maximal subgroup of S or a conjugacy class of core-free maximal subgroups with a fixed index v_Sle {{,mathrm{l},}}(S)^2, depending only on S. We also prove that the number of subgroups of the outer automorphism group of S is bounded by log ^3{{{,mathrm{l},}}(S)} and {{,mathrm{l},}}(S)^2< |S|.