The index of second center in finitely generated groups with a condition on infinite subsets

Let G be a group. We say that G satisfies the condition T(1) if every infinite set of elements of G contains three distinct elements x , y; z such that [x; y; z] = 1 = [y; z; x] = [z; x; y]. In this paper we show that a finitely generated residually finite (locally graded) group G is in T(1) if and only if G=Z2(G) is finite.