Size of Conjugacy Class is Index of Normalizer

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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 $\left[{G : N_G \left({x}\right)}\right]$.

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.

$\blacksquare$