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

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 $\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{m}(G)$, 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 $G\mathrm{.}$ Evidently, $\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{m}(G)⩽\alpha (G)⩽\beta (G)$. For a simple group $G$ we show that $\beta (G)⩽4$ and $\beta (G)-\alpha (G)⩽1$, and both upper bounds are best possible.