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

Authors : Mourad CHELGHAM, Mohamed KERADA, Lemnouar NOUİ

Pages : 22-30

View : 44 | Download : 7

Publication Date : 2018-06-01

Article Type : Research Paper

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 insert ignore into journalissuearticles values(insert ignore into journalissuearticles values(τN k );τ,∞); insert ignore into journalissuearticles values( respectively insert ignore into journalissuearticles values(insert ignore into journalissuearticles values(τN k );τ,∞); ∗ ); is a τ -group insert ignore into journalissuearticles values(respectively τN c for certain integer c=cinsert ignore into journalissuearticles values(k); ); and deduce that a finitely generated FN-group in the class insert ignore into journalissuearticles values(insert ignore into journalissuearticles values(FN k );F,∞); insert ignore into journalissuearticles values(respectively insert ignore into journalissuearticles values(insert ignore into journalissuearticles values(FN k );F,∞); ∗ ); is  -group insert ignore into journalissuearticles values(respectively FN c for certain integer c=cinsert ignore into journalissuearticles values(k););. Thirdly we prove that a finitely generated NF-group in the class insert ignore into journalissuearticles values(insert ignore into journalissuearticles values(FN k );F,∞); insert ignore into journalissuearticles values( respectively insert ignore into journalissuearticles values(insert ignore into journalissuearticles values(FN k );F,∞); ∗ ); is F-group insert ignore into journalissuearticles values(respectively N c F for certain integer c=cinsert ignore into journalissuearticles values(k););. Finally and particularly, we deduce that a finitely generated FN-group in the class insert ignore into journalissuearticles values(insert ignore into journalissuearticles values(FA);F,∞); insert ignore into journalissuearticles values(respectively insert ignore into journalissuearticles values(insert ignore into journalissuearticles values(FC);F,∞); ∗ , insert ignore into journalissuearticles values(insert ignore into journalissuearticles values(FN ₂ );F,∞); ∗ ); is in the class FA insert ignore into journalissuearticles values(respectively FN ₂ , FN ₃ insert ignore into journalissuearticles values(2); ); .
Keywords : FC group FC, F group τNk, τ group FNk, F group insert ignore into journ

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