Maximal subgroups of odd index in finite groups with simple classical socle

Summary

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.