On minimal non-(Hypercentral-By-Cernikov) Groups

Abstract

If X is a class of groups, then a group is said to be minimal non-X if it is not an X-group, while all its proper subgroups belong to X. The main result of this note is if G is a minimal non-ZAC group , then G is a finitely generated perfect group which has no proper subgroup of finite index and such that G/Frat(G) is an infinite simple group, where ZA ( respectively, C) denotes the class of hypercentral groups, (respectively,  the class of Cernikov groups), and Frat(G) stands for the Frattini subgroup of G.