On the minimal dimension of a finite simple group. (English) Zbl 1477.20028
Summary: 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 \(\mathrm{Mindim}(G)\), 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 \(\mathrm{Mindim}(G) \le 3\) 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, \(\mathrm{Mindim}(G) \le \alpha(G) \le\beta(G)\). For a simple group \(G\) we show that \(\beta(G) \le 4\) and \(\beta(G) - \alpha(G) \le 1\), and both upper bounds are best possible.
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