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

TL;DR

This paper explores how the intersection of $\

Contribution

It provides conditions under which the $\

Findings

Identifies when the $\

Establishes a link between the $\

Answers Shemetkov's question affirmatively under certain conditions.

Abstract

We investigate the influence of the intersection of the $\cal F$-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 $\cal F$ has a property that for any finite group $G$, the ${\cal F}$-hypercentre of $G$ coincides with the intersection of all $\cal F$-maximal subgroups of $G$.