On Torsion and Finite Extension of \(FC\) and \(\tau N_{K}\) Groups in Certain Classes of Finitely Generated Groups
OMS-Vol. 2 (2018), Issue 1, pp. 351–360 Open Access Full-Text PDF
Mourad Chelgham, Mohamed Kerada
Abstract: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.
