Finite core-p p-groups

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 finite core-p p-group G has a normal abelian subgroup whose index in G is at most p(2) if p not equal 2, which is the best possible bound, and at most 2(6) if p = 2. (C) 1997 Academic Press.