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

Abstract

Abstract Let 𝔛 {\mathfrak{X}} be a class of groups. A subgroup U of a group G is called 𝔛 {\mathfrak{X}} -maximal in G provided that (a) U ∈ 𝔛 {U\in\mathfrak{X}} , and (b) if U ≤ V ≤ G {U\leq V\leq G} and V ∈ 𝔛 {V\in\mathfrak{X}} , then U = V {U=V} . A chief factor H / K {H/K} of G is called 𝔛 {\mathfrak{X}} -eccentric in G provided ( H / K ) ⋊ G / C G ⁢ ( H / K ) ∉ 𝔛 {(H/K)\rtimes G/C_{G}(H/K)\not\in\mathfrak{X}} . A group G is called a quasi- 𝔛 {\mathfrak{X}} -group if for every 𝔛 {\mathfrak{X}} -eccentric chief factor H / K {H/K} and every x ∈ G {x\in G} , x induces an inner automorphism on H / K {H/K} . We use 𝔛 * {\mathfrak{X}^{*}} to denote the class of all quasi- 𝔛 {\mathfrak{X}} -groups. In this paper we describe all hereditary saturated formations 𝔉 {\mathfrak{F}} containing all nilpotent groups such that the 𝔉 * {\mathfrak{F}^{*}} -hypercenter of G coincides with the intersection of all 𝔉 * {\mathfrak{F}^{*}} -maximal subgroups of G for every group G.