Finite core-\(p\) \(p\)-groups (Q1355539)

Let \(G\) be a finite group. Given a positive integer \(n\), a group \(G\) is called core-\(n\) if \(|H/H_G|\leq n\) for every subgroup \(H\) of \(G\), where \(H_G\) is the normal core of \(H\) in \(G\). 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 \(p\neq 2\), which is the best possible bound, and at most \(2^6\) if \(p=2\). It is shown also, that for odd \(p\) a group \(G\) of this type is nilpotent of class at most 3 and has an abelian normal subgroup of index at most \(p^5\).