Groups with few non-(Hypercentral-By-Finite) Subgroups

Abstract

In this note we study groups with few non-(hyper central-by-finite)subgroups and we prove if G is a minimal non-ZAF group, then G is a finitelygenerated perfect group which has no proper subgroup of finite index and suchthat G/Frat(G) is an infinite simple group, where ZA ( respectively,F) denotesthe class of hypercentral groups, (respectively,the class of Finite groups), andFrat(G) stands for the Frattini subgroup of G.Moreover, we proved a infinitely generated F-perfect MNZAF-group. Gis MNZA-group if, and only if, G is MNFZA-group if, and only if, G is MNZAF