Maximal subgroups of small index of finite almost simple groups

Abstract

AbstractWe 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 $${{\,\mathrm{l}\,}}(S)$$ l ( S ) of a maximal subgroup of S or a conjugacy class of core-free maximal subgroups with a fixed index $$v_S\le {{\,\mathrm{l}\,}}(S)^2$$ v S ≤ 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 $$\log ^3{{{\,\mathrm{l}\,}}(S)}$$ log 3 l ( S ) and $${{\,\mathrm{l}\,}}(S)^2< |S|$$ l ( S ) 2 < | S | .