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

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 $α(G)$ and $β(G)$. Here $α(G)$ (respectively $β(G)$) 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, ${\rm Mindim}(G) \leqslant α(G) \leqslant β(G)$. For a simple group $G$ we show that $β(G) \leqslant 4$ and $β(G) - α(G) \leqslant 1$, and both upper bounds are best possible.