- Journal of New Theory
- Issue: 23
- On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FN k -groups
On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FN k -groups
Authors : Mourad Chelgham, Mohamed Kerada, Lemnouar Noui
Pages : 22-30
View : 11 | Download : 7
Publication Date : 2018-06-01
Article Type : Research
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 c F 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,∞) ∗ , ((FN ₂ )F,∞) ∗ ) is in the class FA (respectively FN ₂ , FN ₃ (2) ) .Keywords : FC-group, (FC)F-group, (τNk)τ-group, (FNk)F-group, ((FNk)F, ∞)-group, ∞)∗-group, finitely generated group