# Prime power order implies not centerless

From Groupprops

## Statement

Any group of prime power order which is nontrivial, has a nontrivial center.

## Related results

- Prime power order implies nilpotent: This follows from the result that a group of prime power order is centerless, the fact that a quotient of a group of prime power order also has prime power order, and by induction.
- Prime power order implies center is normality-large: This is a stronger version of the result stated on this page; the center is not just nontrivial, it intersects
*every*nontrivial normal subgroup, nontrivially. - Locally finite Artinian p-group implies not centerless: This is an attempt to weaken the hypothesis from finiteness, to weaker conditions.

## Proof

The key ingredient for the proof is to consider the action of the group on itself by conjugation (i.e. inner automorphisms) and use the class equation to show that:

Since both are groups of prime power order, the group being nontrivial is equivalent to the center being nontrivial.