Finite groups, minimal bases and the intersection number

Let G $G$ be a finite group and recall that the Frattini subgroup Frat ( ) ${\rm Frat}(G)$ is intersection of all maximal subgroups . In this paper, we investigate number , denoted α $\alpha (G)$ which minimal whose coincides with earlier work, studied in special case where simple here extend analysis to almost groups. particular, prove ⩽ 4 (G) \leqslant 4$ for every best possible. We also establish new results on arbitrary groups, obtaining upper bounds are defined terms chief factors group. Finally, groups present possible related invariant β $\beta call base setting, size as range over faithful primitive actions bound optimal. Along way, study bases action symmetric S b $S_{ab}$ set partitions [ 1 ] $[1,ab]$ into $a$ parts $b$ determining exact ⩾ $a \geqslant b$ This extends work Benbenishty, Cohen Niemeyer.