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.

Proof sketch: By the class equation, $|G| = |Z(G)| + \sum_i [G:C_G(x_i)],$ where each $x_i$ represents a non-central conjugacy class , and $C_G(x_i)$ is the centralizer of $x_i$. For $x_i\notin Z(G)$, the index $[G:C_G(x_i)]$ is a power of $p$ strictly larger than $1$, hence divisible by $p$. Since $|G|$ is divisible by $p$, it follows that $|Z(G)|$ is divisible by $p$, so $Z(G)\neq\{e\}$.