Conjugacy Class of Element of Center is Singleton

Theorems
Let $G$ be a group.

Let $Z \left({G}\right)$ be the center of $G$.

The elements of $Z \left({G}\right)$ form singleton conjugacy classes, and the elements of $G \setminus Z \left({G}\right)$ belong to multi-element conjugacy classes.

Corollary
The number of single-element conjugacy classes of $G$ is the order of $Z \left({G}\right)$ and divides $G$.

Proof
Let $\mathrm C_a$ be the conjugacy class of $a$ in $G$.

Proof of Corollary
Follows trivially from the main result.

Each of the singleton conjugacy classes consists of one of the elements of $Z \left({G}\right)$.

As the Center is a Normal Subgroup, it follows from Lagrange's Theorem that the number of these divides the order of $G$.