On the minimal dimension of a finite simple group (with an appendix by T.C. Burness and R.M. Guralnick)

TL;DR

This paper establishes that the minimal dimension of all finite simple groups is at most 3, computes exact values for non-classical groups, and introduces related invariants with tight bounds.

Contribution

It proves the bound ${\rm Mindim}(G) \leq 3$ for all finite simple groups and determines exact values for non-classical groups, also analyzing related invariants ${\alpha}(G)$ and ${\beta}(G)$.

Findings

${\rm Mindim}(G) \leq 3$ for all finite simple groups

Exact values of ${\rm Mindim}(G)$ for non-classical simple groups

Bounds on ${\beta}(G)$ and ${\beta}(G)-{\alpha}(G)$ are tight

Abstract

Let $G$ be a finite group and let $\mathcal{M}$ be a set of maximal subgroups of $G$. We say that $\mathcal{M}$ is irredundant if the intersection of the subgroups in $\mathcal{M}$ is not equal to the intersection of any proper subset. The minimal dimension of $G$, denoted ${\rm Mindim}(G)$, is the minimal size of a maximal irredundant set of maximal subgroups of $G$. This invariant was recently introduced by Garonzi and Lucchini and they computed the minimal dimension of the alternating groups. In this paper, we prove that ${\rm Mindim}(G) \leqslant 3$ for all finite simple groups, which is best possible, and we compute the exact value for all non-classical simple groups. We also introduce and study two closely related invariants denoted by $\alpha(G)$ and $\beta(G)$. Here $\alpha(G)$ (respectively $\beta(G)$) is the minimal size of a set of maximal subgroups (respectively, conjugate maximal subgroups) of $G$ whose intersection coincides with the Frattini subgroup of $G.$ Evidently, ${\rm Mindim}(G) \leqslant \alpha(G) \leqslant \beta(G)$. For a simple group $G$ we show that $\beta(G) \leqslant 4$ and $\beta(G) - \alpha(G) \leqslant 1$, and both upper bounds are best possible.