On the intersection of the F-maximal subgroups and the generalized F-hypercentre of a finite group

Let F be a class of groups. A chief factor H/K of a group G is called F-central in G provided (H/K)⋊(G/CG(H/K))∈F. We write ZπF(G) to denote the product of all normal subgroups of G whose G-chief factors of order divisible by at least one prime in π are F-central. We call ZπF(G) the πF-hypercentre of G. A subgroup U of a group G is called F-maximal in G provided that (a) U∈F, and (b) if U⩽V⩽G and V∈F, then U=V. In this paper we study the properties of the intersection of all F-maximal subgroups of a finite group. In particular, we analyze the condition under which ZπF(G) coincides with the intersection of all F-maximal subgroups of G.