Prime power order implies not centerless

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 nontrivial group of prime power order is not 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:

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

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 $p$.