Let $H\leq G$ and $N\unlhd G$ such that $G=HN$ and $H\cap N= (1)$.

My question is

> 
> Prove that if two elements of $H$ are conjugate in $G$, then they are conjugate in $H$.
> 
> 
> 

What i have done so far is :

Suppose $a,b\in H$ are conjugate, i.e., we have $g\in G$ such that $a=gbg^{-1}$.

$g\in G$ so, we have $g=mn$ for $m\in H$ and $n\in N$

As $a\in H$ and $a=gbg^{-1}$ we see that $gbg^{-1}\in H$

please provide some hint for this.

Thank You.
$mnbn^{-1}m^{-1}\in H$ this imply $nbn^{-1}\in H$ as $m\in H$

with $nbn^{-1}\in H$ some how i want to conclude $n=e$

I have not yet used the condition that $N\unlhd G$

I thought of using that for $nbn^{-1}$ but then i realized $n\in N$ not $b$. :(