Hyper–(Abelian–by–finite) groups with many subgroups of finite depth

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

“…k , (respectively F N, N F, τ N ). Other results of this type have been obtained, for example in [8,11,12,13,14,15,16]. In this note, we prove that a finitely generated τ N -group G which is in the class…”

“…k F -groups) and particularly a group G is in the class ((F C)F, ∞) if and only if, it is F A-group. On the other hand, in 2005, Trabelsi in [10] (respectively in 2007, Rouabehi and Trabelsi in [17]) proved that a finitely generated soluble group in the class (CN, ∞) * where C is the class of cernikov group (respectively in the class in the class (τ N, ∞) * ) is F N -group (respectively τ N -group) and in 2007 too, Guerbi and Rouabhi in [14] proved that a finitely generated Hyper(abelian-by-finite) group in the class (Ω, ∞) * where Ω is the class of finite depth group, is F N -group. In this paper, we prove that a finitely generated τ N -group in the class ((F N k )τ, ∞) * is in the class τ N c for certain integer c and deduce that a finitely generated F N -group (respectively N F -group) G in the class ((F N k )F, ∞) * ) is in the class F N c (respectively N c F ).…”