# Prime power order implies not centerless

## Statement

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

## 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:

$|G| \equiv |Z(G)| \mod p$

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