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

#### Mourad Chelgham [1] , Mohamed Kerada [2] , Lemnouar Noui [3]

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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,∞), ((FN)F,∞)) is in the class FA (respectively FN, FN(2)).

FC-group, (FC)F-group, (τNk)τ-group, (FNk)F-group, ((FNk)F, ∞)-group, ∞)∗-group, finitely generated group
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Birincil Dil en Araştırma Makalesi Yazar: Mourad Chelgham Yazar: Mohamed Kerada Yazar: Lemnouar Noui
 Bibtex @araştırma makalesi { jnt431047, journal = {Journal of New Theory}, issn = {}, eissn = {2149-1402}, address = {Gaziosmanpaşa Üniversitesi}, year = {2018}, volume = {}, pages = {22 - 30}, doi = {}, title = {On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups}, key = {cite}, author = {Chelgham, Mourad and Kerada, Mohamed and Noui, Lemnouar} } APA Chelgham, M , Kerada, M , Noui, L . (2018). On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups. Journal of New Theory, (23), 22-30. Retrieved from http://dergipark.gov.tr/jnt/issue/37237/431047 MLA Chelgham, M , Kerada, M , Noui, L . "On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups". Journal of New Theory (2018): 22-30 Chicago Chelgham, M , Kerada, M , Noui, L . "On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups". Journal of New Theory (2018): 22-30 RIS TY - JOUR T1 - On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups AU - Mourad Chelgham , Mohamed Kerada , Lemnouar Noui Y1 - 2018 PY - 2018 N1 - DO - T2 - Journal of New Theory JF - Journal JO - JOR SP - 22 EP - 30 VL - IS - 23 SN - -2149-1402 M3 - UR - Y2 - 2018 ER - EndNote %0 Journal of New Theory On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups %A Mourad Chelgham , Mohamed Kerada , Lemnouar Noui %T On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups %D 2018 %J Journal of New Theory %P -2149-1402 %V %N 23 %R %U ISNAD Chelgham, Mourad , Kerada, Mohamed , Noui, Lemnouar . "On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FNk-groups". Journal of New Theory / 23 (Haziran 2018): 22-30.