Journal of New Theory (Jun 2018)
On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups
Abstract
Let k>0 an integer. F, τ,N, Nk, and A denote, respectively, the classes offinite, torsion, nilpotent, nilpotent of class at most k, group in which everytwo generator subgroup is in Nk and abelian groups. The main resultsof this paper is, firstly, to prove that in the class of finitely generatedFN-group, the property FC is closed under finite extension. Secondly, we provethat a finitely generated τN-group in the class ((τNk)τ,∞) ( respectively((τNk)τ,∞)∗)is a τ-group (respectively τNcfor certain integer c=c(k) ) and deduce that a finitely generated FN-group inthe class ((FNk)F,∞) (respectively ((FNk)F,∞)∗)is -group (respectively FNcfor certain integer c=c(k)). Thirdly we prove that a finitely generatedNF-group in the class ((FNk)F,∞) ( respectively ((FNk)F,∞)∗)is F-group (respectively NcFfor certain integer c=c(k)). Finally and particularly, we deduce that afinitely generated FN-group in the class ((FA)F,∞) (respectively ((FC)F,∞)∗,((FN₂)F,∞)∗)is in the class FA (respectively FN₂,FN₃(2)).
