Size of Conjugacy Class is Index of Normalizer

Theorem
Let $$G$$ be a group.

Let $$x \in G$$.

Let $$\mathrm{C}_x$$ be the conjugacy class of $$x$$ in $$G$$.

Let $$N_G \left({x}\right)$$ be the normalizer of $x$ in $G$.

Let $$\left[{G : N_G \left({x}\right)}\right]$$ is the index of $N_G \left({x}\right)$ in $G$.

The number of elements in $$\mathrm{C}_x$$ is $$\left[{G : N_G \left({x}\right)}\right]$$.

Proof
The number of elements in $$\mathrm{C}_x$$ is the number of conjugates of the set $$\left\{{x}\right\}$$.

From Number of Distinct Conjugate Subsets, the number of distinct subsets of a $$G$$ which are conjugates of $$S \subseteq G$$ is $$\left[{G : N_G \left({S}\right)}\right]$$.

The result follows.