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 $$G - Z \left({G}\right)$$ 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 $$G$$.

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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 $$G$$.