# Characteristic rank one implies cyclic-center

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Revision as of 20:17, 13 July 2008 by Vipul (talk | contribs) (New page: ==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-c...)

## 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.