On a question of L.A. Shemetkov concerning the intersection of $\mathcal{F}$-maximal subgroups of finite groups

Abstract

We investigate the influence of the intersection of the $\mathcal{F}$-maximal subgroups on the structure of a finite group. In particular, answering a question of L.A Shemetkov we give conditions under which a hereditary saturated formation $\mathcal{F}$ has a property that for any finite group $G$, the $\mathcal{F}$-hypercentre of $G$ coincides with the intersection of all $\mathcal{F}$-maximal subgroups of $G$.