Finite p-group of normal rank one

Definition
A finite $$p$$-group (i.e., a group of prime power order where the underlying prime is $$p$$) is said to have normal rank one if it satisfies the following equivalent conditions:


 * Every elementary Abelian normal subgroup is cyclic, of order $$p$$
 * Every Abelian normal subgroup is cyclic

Stronger properties

 * Finite p-group of rank one

Weaker properties

 * Finite p-group of characteristic rank one