We prove in this paper that a finitely generated soluble group in which every infinite subset
contains a pair of distinct elements x, y such that〈 x, y〉 is torsion-by-nilpotent …

The purpose of this paper is to present a comprehensive overview of known results on
certain combinatorial problems which are related to infinite subsets of a group. Moreover, in …

The main result of this note is that a finitely generated hyper-(Abelian-byfinite) group G is
finite-by-nilpotent if and only if every infinite subset contains two distinct elements x, y such …

Our main result states that if $ G $ is a finitely generated soluble group having a normal
Abelian subgroup $ A $, such that $ G/A $ and $\left\langle x, a\right\rangle $ are nilpotent …

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 …

Let k> 0 an integer. F, τ, N, Nk, N(2) k and A denote the classes of finite, torsion, nilpotent,
nilpotent of class at most k, group which every two generator subgroup is Nk and abelian …

For any integer n 0, 1, a group G is said to be n-abelian if it satisfies the identity (xy)" x" y".
More generally, G is called an Alperin group if it is n-abelian for some n 0, 1. We consider …

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 …

Let F, N, A and N 2 denote the properties of being finite, nilpotent, abelian and nilpotent of
classes at most 2, respectively. Firstly we consider the class of finitely generated FN-groups …

Let k> 0 an integer. F nilpotent, nilpotent of class at m groups. The main results of this p
property FC is closed under finit class ((τNk) τ,∞)(respectively ((τ deduce that a finitely …