On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FN_k-groups

Abstract

Let k>0 an integer. F, τ, N, Nk, and A denote, respectively, the classes of finite, torsion, nilpotent, nilpotent of class at most k, group in which every two generator subgroup is in Nk and abelian groups. The main results of this paper is, firstly, to prove that in the class of finitely generated FN-group, the property FC is closed under finite extension. Secondly, we prove that a finitely generated τN-group in the class ((τNk)τ,∞) ( respectively ((τNk)τ,∞)∗) is a τ-group (respectively τNc for certain integer c=c(k) ) and deduce that a finitely generated FN-group in the class ((FNk)F,∞) (respectively ((FNk)F,∞)∗) is -group (respectively FNc for certain integer c=c(k)). Thirdly we prove that a finitely generated NF-group in the class ((FNk)F,∞) ( respectively ((FNk)F,∞)∗) is F-group (respectively NcF for certain integer c=c(k)). Finally and particularly, we deduce that a finitely generated FN-group in the class ((FA)F,∞) (respectively ((FC)F,∞)∗, ((FN₂)F,∞)∗) is in the class FA (respectively FN₂, FN₃(2)).