Maximal irredundant families of minimal size in the alternating group

TL;DR

This paper determines the minimal size of maximal irredundant families of maximal subgroups in the alternating group on n letters, providing new insights into subgroup intersection properties in finite groups.

Contribution

It explicitly computes the value of Mindim(G) for the alternating group, advancing understanding of subgroup structures in finite simple groups.

Findings

Computed Mindim(G) for all alternating groups on n letters.

Established the structure of maximal irredundant families in these groups.

Provided formulas or bounds for the minimal size in specific cases.

Abstract

Let $G$ be a finite group. A family $\mathcal{M}$ of maximal subgroups of $G$ is called `irredundant' if its intersection is not equal to the intersection of any proper subfamily. $\mathcal{M}$ is called `maximal irredundant' if $\mathcal{M}$ is irredundant and it is not properly contained in any other irredundant family. We denote by $\mbox{Mindim}(G)$ the minimal size of a maximal irredundant family of $G$. In this paper we compute $\mbox{Mindim}(G)$ when $G$ is the alternating group on $n$ letters.