Locally graded groups with a nilpotency condition on infinite subsets

A group G is locally graded if every finitely generated nontrivial subgroup of G has a nontrivial finite image. Let N (2, k)* denote the class of grou ps in which every infinite subset contains a pair of elements that generate a nilpotent subgroup of class at most k. We show that if G is a finitely g enerated locally graded N (2, k)*-group, then there is a positive integer c depending only on k such that G/Z(c)(G) is finite.