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, N_k, 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 N_k 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 ((τN_k)τ,∞) ( respectively ((τN_k)τ,∞)^∗) is a τ-group (respectively τN_c for certain integer c=c(k) ) and deduce that a finitely generated FN-group in the class ((FN_k)F,∞) (respectively ((FN_k)F,∞)^∗) is -group (respectively FN_c for certain integer c=c(k)). Thirdly we prove that a finitely generated NF-group in the class ((FN_k)F,∞) ( respectively ((FN_k)F,∞)^∗) is F-group (respectively N_cF 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⁾).

Keywords

• FC-group,(FC)F-group,(τNk)τ-group,(FNk)F-group,((FNk)F,∞)-group,∞)∗-group,finitely generated group

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