Finite core- p p -groups

Forna positive integer, a groupGis calledcore-nifH/HGhas order at mostnfor every subgroupHofG(whereHGis the normal core ofH, the largest normal subgroup ofGcontained inH). It is proved that a finite core-pp-groupGhas a normal abelian subgroup whose index inGis at mostp2ifp≠2, which is the best possible bound, and at most 26ifp=2.