Locally nilpotent groups with all subgroups normal-by-(finite rank)

Let G be a locally nilpotent group in which H/Core(G)H has finite rank for all subgroups H. If G is torsion-free then G/Z(G) has finite rank. In general G has an abelian normal subgroup A with G/A of finite rank, and H/Core(G)H has bounded rank for all H. Further results are obtained in the case where H/Core(G)H has rank at most t (a fixed positive integer) for all subgroups H.