Asadollah Faramarzi Salles

Let G be a group, we say that G satisfies the property T (∞) provided that, every infinite set of elements of G contains elements x ≠ y, z such that [x, y, z] = 1 = [y, z, x] = [z, x, y]. We denote by C the class of all polycyclic groups, S the class of all soluble groups, R the class of all residually finite groups, L the class of all locally graded groups, N2 the class of all nilpotent group of class at most two, and F the class of all finite groups. In this paper, first we shall prove that if G is a finitely generated locally graded group, then G satisfies T (∞) if and only if G/Z2 (G) is finite, and then we shall conclude that if G is a finitely generated group in T (∞), then G ∈ L ⇔ G ∈ R ⇔ G ∈ S ⇔ G ∈ C ⇔ G ∈ N2 F. © 2018 University of Isfahan.

A group G is called n-centralizer if it has n distinct centralizers. In this paper, in analogs to n-centralizer, we say a group G is n-exterior centralizer provided G has n distinct exterior centralizers. The current paper is devoted to characterize all groups that are n-exterior centralizer, where n ε {1,2,3,4,5}. © 2018, © 2018 Taylor & Francis Group, LLC.

Let n 0, 1 be an integer and cny NLD{B}-n$ be the variety of n-Bell groups defined by the law [xn,y][x,yn]-1 = 1. Let cny NLD{B}-n$ be the class of groups in which for any infinite subsets X and Y there exist x X and y Y such that [xn,y][x,yn]-1 = 1. In this paper we prove cny NLDB-ncap L=(B-ncup F L$, where cny NLDF}$ and cny NLDL$ are the classes of all finite groups and all locally graded groups, respectively. © 2016 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Suzhou University.

Let G be an infinite group and n ∈ {3, 6}∪{2k|k ∈ N}. In this paper, we prove that G is an n-Kappe group if and only if for any two infinite subsets X and Y of G, there exist x ∈ X and y ∈ Y such that [xn, y, y] = 1. © 2013 University of Isfahan.

Let G be a group. We say that G ∈ T(∞) provided that every infinite set of elements of G contains three distinct elements x,y,z such that x≠y,[x,y,z]=1=[y,z,x]=[z,x,y]. We use this to show that for a finitely generated soluble group G, G/Z 2(G) is finite if and only if G ∈ T(∞). © 2012 Australian Mathematical Publishing Association Inc.