Abstract :
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.
