Intersection of Conjugacy Class and Subgroup

Proposition

Let $G$ be a finite group, and let $H⩽G$ be a subgroup. Define $Cl(x)$ as the conjugacy class containing $x$. Consider the group action of $G$ on the set of right cosets $\Omega =\{Hg:g\in G\}$ and the set $\mathrm{fix}(x)=\{Hg:Hgx=Hg\}$. Then

$$|Cl(x)|=\frac{[G:H]|Cl(x)\cap H|}{|\mathrm{fix}(x)|}.$$

\begin{proof} Define $S=\{g\in G:gx{g}^{-1}\in H\}$, then we can compute that:

• $|S|=|H\cap Cl(x)||C_G(x)|=|H\cap Cl(x)||G|/|Cl(x)|$;
• $|S|=|\mathrm{fix}(x)||H|$.

Now we finish the proof. \end{proof}