Locally graded groups with a condition on infinite subsets

Let $G$ be a groupý, ýwe say that $G$ satisfies the property $\mathcal{T}(\infty)$ provided thatý, ýevery infinite set of elements of $G$ contains elements $x\neq yý, ýz$ such that $[xý, ýyý, ýz]=1=[yý, ýzý, ýx]=[zý, ýxý, ýy]$ý.
ýWe denote by $\mathcal{C}$ the class of all polycyclic groupsý, ý$\mathcal{S}$ the class of all soluble groupsý, ý$\mathcal{R}$ the class of all residually finite groupsý, ý$\mathcal{L}$ the class of all locally graded groupsý, ý$\mathcal{N}_2$ the class of all nilpotent group of class at most twoý, ýand $\mathcal{F}$ the class of all finite groupsý. ýIn this paperý, ýfirst we shall prove that if $G$ is a finitely generated locally graded groupý, ýthen $G$ satisfies $\mathcal{T}(\infty)$ if and only if $G/Z_2(G)$ is finiteý, ýand then we shall conclude that if $G$ is a finitely generated group in $\mathcal{T}(\infty)$ý, ýthený ý\[G\in\mathcal{L}\Leftrightarrow G\in\mathcal{R}\Leftrightarrow G\in\mathcal{S}\Leftrightarrow G\in\mathcal{C}\Leftrightarrow G\in\mathcal{N}_2\mathcal{F}.\]ý