Let G be a non-abelian group and let Z (G) be the center of G. Associate a graph ΓG (called
non-commuting graph of G) with G as follows: Take G\Z (G) as the vertices of ΓG and join two …

The central concept in this monograph is that of a soluble group-a group which is built up
from abelian groups by repeatedly forming group extensions. It covers all the major areas …

A condition on finitely generated soluble groups Page 1 COMMUNICATIONS IN ALGEBRA,
27(1 I), 5633-5638 (1 999) A CONDITION ON FINITELY GENERATED SOLUBLE GROUPS …

In this note, we prove that, in every finitely generated soluble group G, G/Z2 (G) is finite if and
only if in every infinite subset X of G there exist different x, y such that [x, y, y= 1. BH …

Let k be a positive integer. We denote by ɛk (∞) the class of all groups in which every
infinite subset contains two distinct elements x, y such that [x, ky]= 1. We say that a group G …

Let $ n> 0$ be an integer and $\mathcal {X} $ be a class of groups. We say that a group $ G
$ satisfies the condition $(\mathcal {X}, n) $ whenever in every subset with $ n+ 1$ elements …

A group G is said to be in E_k^* (ka positive integer), if every infinite subset of G contains a
pair of elements that generate ak-Engel group.¶ It is shown that a finitely generated locally …

A group G is locally graded if every finitely generated nontrivial subgroup of G has a
nontrivial finite image. Let N (2, k)* denote the class of groups in which every infinite subset …

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 …

ON RESIDUALLY FINITE GROUPS WITH AN ENGEL CONDITION ON INFINITE SUBSETS
Let k be a positive integer. We denote by E k(∞) the c Page 1 Houston Journal of Mathematics …