A nontrivial finite p-group has nontrivial centre

Statement

Let $p$ be a prime number and let $G$ be a finite p-group with $G \neq \{e\}$. Then the centre $Z(G)$ is not $\{e\}$.

Remarks

In the class equation $|G| = |Z(G)| + \sum [G : C_G(a_i)]$, the left side and each term of the sum are divisible by $p$. So $p$ divides $|Z(G)|$, and $|Z(G)| \geq p$. Equivalently, apply the fixed point lemma to the action of $G$ on itself by conjugation. A consequence is that each finite $p$-group is nilpotent.

Explanations

L1 human

The conjugacy classes of a $p$-group that have more than one element all have sizes divisible by $p$. The order of the group is also divisible by $p$. So the classes with one element, which form the centre, cannot be just the identity.