On classes of finite groups with simple non-abelian chief factors

Abstract

Let J be a class of non-abelian simple groups and X be a class of groups. A chief factor H/K of a group G is called X-central in G provided (H/K) ⋊ G/CG(H/K) ∈ X. We say that G is a Jcs-X-group if every chief X-factor of G is X-central and other chief factors of G are simple J-groups. We use XJcs to denote the class of all Jcs-X-groups. A subgroup U of a group G is called X-maximal in G provided that (a) U ∈ X, and (b) if U ≤ V ≤ G and V ∈ X, then U = V . In this paper we described the structure of Jcs-H-groups for a solubly saturated formation H and all hereditary saturated formations F containing all nilpotent groups such that the FJcs-hypercenter of G coincides with the intersection of all FJcs-maximal subgroups of G for every group G.