Prime power order implies not centerless

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Statement

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

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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.