Boolean lattices in finite alternating and symmetric groups

TL;DR

This paper classifies subgroups of finite alternating and symmetric groups whose subgroup lattice is Boolean of rank at least 3, revealing two main types related to stabilizers of partitions and product structures, with applications to conjectures in group theory.

Contribution

It provides a complete classification of Boolean subgroup lattices in finite alternating and symmetric groups, identifying two main structural types and confirming related conjectures.

Findings

Two main types of Boolean lattices identified: stabilizers of chains of regular partitions and product structures.

Classification includes sporadic examples and twisted versions.

Application to a conjecture on Boolean overgroup lattices and related group theory problems.

Abstract

Given a group $G$ and a subgroup $H$, we let $\mathcal{O}_G(H)$ denote the lattice of subgroups of $G$ containing $H$. This paper provides a classification of the subgroups $H$ of $G$ such that $\mathcal{O}_{G}(H)$ is Boolean of rank at least $3$, when $G$ is a finite alternating or symmetric group. Besides some sporadic examples and some twisted versions, there are two different types of such lattices. One type arises by taking stabilizers of chains of regular partitions, and the other type arises by taking stabilizers of chains of regular product structures. As an application, we prove in this case a conjecture on Boolean overgroup lattices, related to the dual Ore's theorem and to a problem of Kenneth Brown.