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

 * Let $$H$$ be a union of conjugacy classes of $$G$$.


 * Let $$H$$ be normal in $$G$$.

Then $$\forall g \in g: g H g^{-1} \subseteq H$$.

So $$\forall x \in H: g x g^{-1} \in H$$ by definition of Conjugate of a Set.

Thus $$x$$ is in a conjugacy class of $$G$$.