A relation between infinite subsets and exterior center in groups

Let G be a group. Neumann to answer a question of Paul Erdos proved that every infinite subset of G has two different comuting elements  if and only if G is center-by-finite. In this paper, we deal with Erdoschr('39')s question in different aspect and we show that every infinite subset X of G has two different elements x and y such that x^y=1  if and only if the exterior  center of G ihas finite index.