Prime power order implies not centerless

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

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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 order a power of p, the group being nontrivial is equivalent to the center being nontrivial -- either means that the two sides of the congruence are 0 mod p.