Finite groups with the same join graph as a finite nilpotent group

Given a finite group $G,$ we denote by $\Delta(G)$ the graph whose vertices are the proper subgroups of $G$ and in which two vertices $H$ and $K$ are joined by an edge if and only if $G=\langle H,K\rangle.$ We prove that if there exists a finite nilpotent group $X$ with $\Delta(G)\cong \Delta(X),$ then $G$ is supersoluble

On the minimal dimension of a finite simple group

Let G be a finite group and let M be a set of maximal subgroups of G. We say that M is irredundant if the intersection of the subgroups in M is not equal to the intersection of any proper subset. The minimal dimension of G, denoted 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 Mindim(G)\u2a7d3 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 \u3b1(G) and \u3b2(G). Here \u3b1(G) (respectively \u3b2(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, Mindim(G)\u2a7d\u3b1(G)\u2a7d\u3b2(G). For a simple group G we show that \u3b2(G)\u2a7d4 and \u3b2(G) 12\u3b1(G)\u2a7d1, and both upper bounds are best possible