A subgroup $H$ is normal in $G$ iff $H$ is the union of some conjugacy classes of $G$ How to show that:

A subgroup $H$ is normal in $G$ iff $H$ is the union of some conjugacy classes of $G$.

 A: Hints:
Suppose
$$H\lhd G\implies \forall\,h\in H\;\wedge\;\forall\,g\in G\,,\;\;h^g:=g^{-1}hg\in H$$
The other direction is almost trivial: denote by $\,[g]\;$ the conjugation class of an element $\,g\in G\,$ , then
$$\forall\,h\in H\;,\;g\in G\;,\;\;h^g\in[h]\subset H\;\ldots\ldots$$
A: Hint: Look at the group action of G on itself by conjugation.
A: Consider the  relation on $G$ given by
$$
a \sim b \qquad\text{if and only if}\qquad \text{there exists $g \in G$ such that $b = g^{-1} a g$},
$$
so this is the conjugacy relation. It is clearly an equivalence relation, so one may consider the conjugacy classes $[a] = \{ g^{-1} a g : g \in G \}$, which then form a partition of $G$.
Now by definition a subgroup $H$ is normal in $G$ if and only if 

if $a \in H$ and $b \sim a$, then $b \in H$,

or in other words, 

if $a \in H$, then $[a] \subseteq H$. 

If $a \in H$, we have then $a \in \{ a \} \subseteq [a] \subseteq H$. It follows that
$$
H = \bigcup_{a \in H} \{ a \} = \bigcup_{a \in H} [a].
$$
