Conjugacy Class of Element of Center is Singleton/Corollary

Corollary to Conjugacy Class of Element of Center is Singleton
Let $G$ be a group.

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

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

Proof
From Conjugacy Class of Element of Center is Singleton, each of the singleton conjugacy classes consists of one of the elements of $\map Z G$.

By Center of Group is Subgroup, $\map Z G$ is a subgroup of $G$.

It follows from Lagrange's Theorem that the number of these divides the order of $G$.