Finite groups with only 𝔉 -normal and 𝔉 -abnormal subgroups

Abstract

Let G be a finite group, and let 𝔉 be a class of groups. A chief factor H / K of G is said to be 𝔉-central (in G) if the semidirect product (H / K) ⋊ (G / Cɢ (H / K)) ∈ 𝔉. We say that a subgroup A of G is 𝔉-normal in G if every chief factor H / K of G between Aɢ and Aᴳ is 𝔉-central in G and 𝔉-abnormal in G if V is not 𝔉-normal in W for every two subgroups V < W of G such that A ≤ V. We give a description of finite groups in which every subgroup is either 𝔉-normal or 𝔉-abnormal.