Size of Conjugacy Class is Index of Normalizer

Theorem
Let $G$ be a group.

Let $x \in G$.

Let $\operatorname C_x$ be the conjugacy class of $x$ in $G$.

Let $\map {N_G} x$ be the normalizer of $x$ in $G$.

Let $\index G {\map {N_G} x}$ be the index of $\map {N_G} x$ in $G$.

The number of elements in $\operatorname C_x$ is $\index G {\map {N_G} x}$.

Proof
The number of elements in $\operatorname C_x$ is the number of conjugates of the set $\set x$.

From Number of Distinct Conjugate Subsets is Index of Normalizer, the number of distinct subsets of a $G$ which are conjugates of $S \subseteq G$ is $\index G {\map {N_G} S}$.

The result follows.