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• W2922884512 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." @default.
• W2922884512 created "2019-04-01" @default.
• W2922884512 creator A5024944956 @default.
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• W2922884512 date "2019-03-22" @default.
• W2922884512 modified "2023-09-27" @default.
• W2922884512 title "On the minimal dimension of a finite simple group (with an appendix by T.C. Burness and R.M. Guralnick)" @default.
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