Abstract

We discuss the completion of the classification of maximal subgroups of odd index in finite groups with simple classical socle.

Introduction

The subgroup of a finite group G generated by the set of all its minimal non-trivial normal subgroups is called the socle of G and is denoted by Soc(G). A finite group is almost simple if its socle is a nonabelian simple group. It is well known that a finite group G is almost simple if and only if there exists a nonabelian finite simple group L such that L ≃ Inn(L) ⊴ G ≤ Aut(L). In this case Inn(L) = Soc(G). One of the greatest results in the theory of finite permutation groups was obtained by Liebeck and Saxl [7] and independently by Kantor [3]. They gave the classification of finite primitive permutation groups of odd degree. In particular, for each finite group G whose socle is a simple classical group they specified types of subgroups which can be maximal subgroups of odd index in G. However, not every subgroup of these types is a maximal subgroup of odd index in G. Thus, the classification of maximal subgroups of odd index in finite groups with a simple classical socle is not complete. In this paper, we discuss the completion of the classification of maximal subgroups of odd index in finite groups with simple classical socle.