Maximal subgroups of small index of finite almost simple groups (Q2081236)

All groups under consideration are finite. An almost simple group \(R\) is a subgroup of \(\operatorname{Aut}(S)\) for some simple group \(S\), such that \(S\leq R\). If \(S\) is a non-abelian simple group, then the socle of \(R\) is \(S\). Let \(l(X)\) be the smallest index of a core-free subgroup of a group \(X\). In the paper under review, the authors prove 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 \(l(S)\) of a maximal subgroup of \(S\) or a conjugacy class of core-free maximal subgroups with a fixed index \(v_S\leq l(S)^2\), depending only on \(S\). In addition, it is argued that if \(S\) is a non-abelian simple group, then \(l(S)^2<|S|\) and the number of subgroups of the outer automorphism group of \(S\) is bounded by \(\log_2^3 l(S)\). All results largely depend on the classification of finite simple groups.