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.