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

Abstract

Let Ӿ be a class of groups. A subgroup U of a group G is called Ӿ -maximal in G provided that (a) U ∊ Ӿ , and (b) if U ≤ V ≤ G and V ∊ Ӿ, then U = V. A chief factor H / K of G is called Ӿ-eccentric in G provided (H / K) ⋊ G / Cɢ (H / K) ∉ Ӿ. A group G is called a quasi-Ӿ-group if for every Ӿ-eccentric chief factor H / K and every x ∊ G, x induces an inner automorphism on H / K. We use Ӿ* to denote the class of all quasi Ӿ-groups. In this paper we describe all hereditary saturated formations 𝔉 containing all nilpotent groups such that the 𝔉* -hypercenter of G coincides with the intersection of all 𝔉* -maximal subgroups of G for every group G.