# Conjugacy Class of Element of Center is Singleton

## Theorem

Let $G$ be a group.

Let $\map Z G$ denote the center of $G$.

The elements of $\map Z G$ form singleton conjugacy classes, and the elements of $G \setminus \map Z G$ belong to multi-element conjugacy classes.

### Corollary

The number of single-element conjugacy classes of $G$ is the order of $\map Z G$ and divides $\order G$.

## Proof

Let $\conjclass a$ be the conjugacy class of $a$ in $G$.

 $\ds a$ $\in$ $\ds \map Z G$ $\ds \leadstoandfrom \ \$ $\ds \forall x \in G: \,$ $\ds x a$ $=$ $\ds a x$ $\ds \leadstoandfrom \ \$ $\ds \forall x \in G: \,$ $\ds x a x^{-1}$ $=$ $\ds a$ $\ds \leadstoandfrom \ \$ $\ds \conjclass a$ $=$ $\ds \set a$

$\blacksquare$