Size of the orbits of a normal subgroup

Question

So this is the question:

Let $H$ be a finite subgroup of $G$, and let $(h,h')(x)=hxh^{-1}$ define an achtion of $H\times H$ on $G$, prove that $H$ is a normal subgroup of $G$ if and only if every orbit of this action contains $|H|$ elements.

I really have no clue where to start...

Answer 1

I will show the one direction and the other is an exercises for you.

Note that for a normal group $N$, $gN=Ng$ thus for $gn$ there is a uniqe $n_2$ such that $gn=n_2g$.

Let $H$ be a normal subgroup of $G$ and $K=H\times H$

$|K:Stab(x)|$ is the size of orbit containing $x$.

$$hxh'^{-1}=x$$ $$xh_2h'^{-1}=x$$ $$h_2=h'$$ Note that for any $h$, there is a uniqe such $h_2$. And as we see $h_2$ uniqly determine the $h'$.

Thus, $Stab(x)=\{(h,h')|hxh'^{-1}=x \}$ has exactly $|H|$ elements. So, $|K:Stab(x)|=|H|$.