On Torsion and Finite Extension of $FC$ and $\tau N_{K}$ Groups in Certain Classes of Finitely Generated Groups

• Main Authors: Mourad Chelgham, Mohamed Kerada
• Format: Article
• Language: English
• Published: Ptolemy Scientific Research Press 2018-11-01
• Series: Open Journal of Mathematical Sciences
• Subjects: \((FC)\tau\)-group; \((FC)F\)-group; \(((\tau N_{k})\tau; \infty)\)-group; \(((FN_{k})F; \infty)\)-group; \(((\tau N_{k})\tau; \infty)^{\ast}\)-group; \(((FN_{k})F; \infty )^{\ast }\)-group.
• Online Access: https://openmathscience.com/on-torsion-and-finite-extension-of-fc-and-tau-n_k-groups-in-certain-classes-of-finitely-generated-groups-2/

Let \(k>0\) an integer. \(F\), \(\tau \), \(N\), \(N_{k}\), \(N_{k}^{(2)}\) and \(A\) denote the classes of finite, torsion, nilpotent, nilpotent of class at most \(k\), group which every two generator subgroup is \(N_{k}\) and abelian groups respectively. The main results of this paper is, firstly, we prove that, in the class of finitely generated \(\tau N\)-group (respectively \(FN\)-group) a \((FC)\tau \)-group (respectively \((FC)F\)-group) is a \(\tau A\)-group (respectively is \(FA\)-group). Secondly, we prove that a finitely generated \(\tau N\)-group (respectively \(FN\)-group) in the class \(((\tau N_{k})\tau ,\infty)\) (respectively \(((FN_{k})F,\infty)\)) is a \(\tau N_{k}^{(2)}\)-group (respectively \(FN_{k}^{(2)}\)-group). Thirdly we prove that a finitely generated \(\tau N\)-group ( respectively \(FN\)-group) in the class \(((\tau N_{k})\tau ,\infty)^{\ast}\) (respectively \(((FN_{k})F,\infty)^{\ast}\)) is a \(\tau N_{c}\)-group (respectively \(FN_{c}\)-group) for certain integer \(c\) and we extend this results to the class of \(NF\)-groups.