Finite p-Group Has Nontrivial Center

Let $p$ be a prime and let $G$ be a finite $p$ -group of order $p^n$ , where $n\ge 1$ . Then the center $Z(G)$ is nontrivial. More precisely,

p\mid |Z(G)|.

Examples

If $G$ is abelian, then $Z(G)=G$ .
For $D_8=\langle r,s\mid r^4=s^2=e,\ srs=r^{-1}\rangle$ , one has $Z(D_8)=\{e,r^2\}$ .

Remarks

This follows from the class equation: every noncentral conjugacy class has cardinality divisible by $p$ , so $|Z(G)|\equiv |G|\equiv 0\pmod p$ .