On Finite Extensions of FC-Groups in the Class of N F-Groups

Abstract

Let F, N, A and N-2 denote the properties of being finite, nilpotent, abelian and nilpotent of classes at most 2, respectively. Firstly we consider the class of finitely generated FN-groups. We show that the property FC is closed under finite extensions, and extend this result to finitely generated NF-groups. Secondly we prove that a finitely generated NF-group G is in the class ((FC) F,infinity) if and only if G is an FA-group. Finally we prove that a finitely generated NF-group in the class ((FC) F,infinity)* is an FN2-group. Moreover, G/Z(2)(G) is finite.