Union of Conjugacy Classes is Normal

Theorem
Let $$G$$ be a group.

Let $$H \le G$$.

Then $$H$$ is normal iff $$H$$ is a union of conjugacy classes of $$G$$.

Proof

 * $$H$$ is normal in $$G$$
 * $$\iff \forall g\in G:gHg^{-1}\subseteq H$$
 * $$\iff \forall x\in H:\forall g\in G:gxg^{-1}\in H$$
 * $$\iff \forall x\in H: H$$ contains the conjugacy class of $$x$$ in $$G$$.
 * $$\iff H$$ is a union of conjugacy classes of G.