Nilpotent p-group

Definition
Let $$p$$ be a prime. A nilpotent p-group is a group satisfying the following equivalent conditions:


 * 1) It is a $$p$$-group (see p-group -- every element has order a power of $$p$$) that is also a nilpotent group.
 * 2) It is a a nilpotent group in which every finitely generated subgroup is a finite p-group.

Facts

 * Every finite $$p$$-group is nilpotent -- see prime power order implies nilpotent.
 * There exist infinite non-nilpotent $$p$$-groups. See p-group not implies nilpotent.