Second maximal subgroups of the finite alternating and symmetric groups

TL;DR

This paper investigates second maximal subgroups in finite alternating and symmetric groups, revealing that, aside from three known cases, such subgroups are contained in at most three maximal subgroups, contributing to understanding subgroup structures.

Contribution

It provides a classification of second maximal subgroups in finite alternating and symmetric groups, identifying bounds on their maximal overgroups, except for three special cases.

Findings

Most second maximal subgroups are contained in no more than three maximal subgroups.

Identifies three exceptional cases with different containment properties.

Advances understanding of subgroup lattice structures in finite simple groups.

Abstract

A subgroup of a finite group G is said to be second maximal if it is maximal in every maximal subgroup of G that contains it. A question which has received considerable attention asks: can every positive integer occur as the number of the maximal subgroups that contain a given second maximal subgroup in some finite group G? Various reduction arguments are available except when G is almost simple. Following the classification of the finite simple groups, finite almost simple groups fall into three categories: alternating and symmetric groups, almost simple groups of Lie type, sporadic groups and automorphism groups of sporadic groups. This thesis investigates the finite alternating and symmetric groups, and finds that in such groups, except three well known examples, no second maximal subgroup can be contained in more than 3 maximal subgroups.