Finite p-Group Has Nontrivial Center

Finite p-Group Has Nontrivial Center: Let $p$ be a prime and let $G$ be a finite p-group , i.e. $|G|=p^n$ for some integer $n\ge 1$. Then the center $Z(G)$ is nontrivial. More precisely,

so in particular $|Z(G)|\ge p$.

This is a direct consequence of the class equation , which decomposes $|G|$ into $|Z(G)|$ plus sizes of non-central conjugacy classes , each of which has cardinality divisible by $p$ in a $p$-group.

Proof sketch: The class equation has the form

where each $C_i$ is a conjugacy class of an element not in $Z(G)$. For a finite $p$-group, each $|C_i|$ is a power of $p$ strictly greater than $1$, hence divisible by $p$. Reducing the equation modulo $p$ gives $|G|\equiv |Z(G)|\pmod p$. Since $p\mid |G|$, it follows that $p\mid |Z(G)|$.

Examples:

• If $G$ is abelian, then $Z(G)=G$, so the conclusion holds trivially (and $|Z(G)|=|G|=p^n$).
• In the dihedral group of order $8$, $D_8=\langle r,s \mid r^4=e,\ s^2=e,\ srs=r^{-1}\rangle$, the center is $Z(D_8)=\{e,r^2\}$, which has size $2$ (and $2\mid 8$).
• If $|G|=p$ then $G$ is cyclic (hence abelian), so again $Z(G)=G\neq \{e\}$.