Locally graded groups with a condition on infinite subsets

Document Type: Research Paper

Authors

Damghan University

Abstract

Let $G$ be a group‎, ‎we say that $G$ satisfies the property $\mathcal{T}(\infty)$ provided that‎, ‎every infinite set of elements of $G$ contains elements $x\neq y‎, ‎z$ such that $[x‎, ‎y‎, ‎z]=1=[y‎, ‎z‎, ‎x]=[z‎, ‎x‎, ‎y]$‎.
‎We denote by $\mathcal{C}$ the class of all polycyclic groups‎, ‎$\mathcal{S}$ the class of all soluble groups‎, ‎$\mathcal{R}$ the class of all residually finite groups‎, ‎$\mathcal{L}$ the class of all locally graded groups‎, ‎$\mathcal{N}_2$ the class of all nilpotent group of class at most two‎, ‎and $\mathcal{F}$ 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 $\mathcal{T}(\infty)$ if and only if $G/Z_2(G)$ is finite‎, ‎and then we shall conclude that if $G$ is a finitely generated group in $\mathcal{T}(\infty)$‎, ‎then‎ ‎\[G\in\mathcal{L}\Leftrightarrow G\in\mathcal{R}\Leftrightarrow G\in\mathcal{S}\Leftrightarrow G\in\mathcal{C}\Leftrightarrow G\in\mathcal{N}_2\mathcal{F}.\]‎

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