Conjugacy Class Size Lemma

Conjugacy Class Size Lemma: Let $G$ be a finite group and let $g\in G$. Let $\mathrm{Cl}(g)$ denote the conjugacy class of $g$, and let $C_G(g)$ denote the centralizer of $g$. Then $|\mathrm{Cl}(g)| = [G : C_G(g)],$ where $[G:C_G(g)]$ is the index of $C_G(g)$ in $G$.

This is an instance of the orbit–stabilizer theorem applied to the conjugation action of $G$ on itself.

Proof sketch: Under conjugation, the orbit of $g$ is $\mathrm{Cl}(g)$. The stabilizer of $g$ is $\{x\in G : xgx^{-1}=g\}=C_G(g)$. Orbit–stabilizer gives $|\mathrm{Cl}(g)|=[G:C_G(g)]$.