Finite groups, minimal bases and the intersection number

TL;DR

This paper studies the intersection number of finite groups, especially almost simple groups, establishing upper bounds and exploring related invariants like the base number, with implications for group actions and subgroup intersections.

Contribution

It extends previous work on the intersection number to almost simple groups, proving the bound $ ext{alpha}(G) \,\leq 4$, and introduces new bounds for arbitrary finite groups based on chief factors.

Findings

$ ext{alpha}(G) \leq 4$ for all almost simple groups, and this bound is optimal.

Established bounds on the base number $eta(G)$ for almost simple groups, with $eta(G) \leq 4$ being optimal.

Determined exact base sizes for symmetric group actions on partitions, extending prior research.

Abstract

Let $G$ be a finite group and recall that the Frattini subgroup ${\rm Frat}(G)$ is the intersection of all the maximal subgroups of $G$. In this paper, we investigate the intersection number of $G$, denoted $\alpha(G)$, which is the minimal number of maximal subgroups whose intersection coincides with ${\rm Frat}(G)$. In earlier work, we studied $\alpha(G)$ in the special case where $G$ is simple and here we extend the analysis to almost simple groups. In particular, we prove that $\alpha(G) \leqslant 4$ for every almost simple group $G$, which is best possible. We also establish new results on the intersection number of arbitrary finite groups, obtaining upper bounds that are defined in terms of the chief factors of the group. Finally, for almost simple groups $G$ we present best possible bounds on a related invariant $\beta(G)$, which we call the base number of $G$. In this setting, $\beta(G)$ is the minimal base size of $G$ as we range over all faithful primitive actions of the group and we prove that the bound $\beta(G) \leqslant 4$ is optimal. Along the way, we study bases for the primitive action of the symmetric group $S_{ab}$ on the set of partitions of $[1,ab]$ into $a$ parts of size $b$, determining the exact base size for $a \geqslant b$. This extends earlier work of Benbenishty, Cohen and Niemeyer.