A p-group has nontrivial center

A p-group has nontrivial center: Let $G$ be a finite p-group , i.e. $|G|=p^n$ for some prime $p$ and integer $n\ge 1$. Then the center $Z(G)$ is nontrivial; in fact, $|Z(G)|$ is divisible by $p$, so $|Z(G)|\ge p$.

This is a standard application of the class equation , which decomposes $|G|$ into the size of the center plus sizes of non-central conjugacy classes.