Characteristic rank one implies cyclic-center

Statement

Property-theoretic statement

The group property of being a finite p-group of characteristic rank one is stronger than the group property of being a cyclic-center group.

Verbal statement

If a group of prime power order has characteristic rank one, then its center is a cyclic group.

Proof

This is a direct application of the fact that the center of a group is an Abelian characteristic subgroup.