Locally graded groups with a condition on infinite subsets

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 $[x‎, ‎y‎, ‎z]=۱=[y‎, ‎z‎, ‎x]=[z‎, ‎x‎, ‎y]$‎. ‎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, ‎$\mathcal{N}_۲$ 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 $G/Z_۲(G)$ is finite‎, ‎and then we shall conclude that if $G$ is a finitely generated group in $\mathcal{T}(\infty)$‎, ‎then‎ ‎\[G\in\mathcal{L}\Leftrightarrow G\in\mathcal{R}\Leftrightarrow G\in\mathcal{S}\Leftrightarrow G\in\mathcal{C}\Leftrightarrow G\in\mathcal{N}_۲\mathcal{F}.\]‎

Keywords:

‎Finitely generated groups‎ , ‎Residually finite groups‎ , ‎Locally graded groups