locally graded groups with a condition on infinite subsets

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∈n2f.