Locally Finite Groups All of Whose Subgroups are Boundedly Finite over Their Cores

For n a positive integer, a group G is called core-n if H/H-G has order at most n for every subgroup H of G (where H-G is the normal core of H, the largest normal subgroup of G contained in H). It is proved that a locally finite core-n group G has an abelian subgroup whose index in G is bounded in terms of n.