$H,K$ are normal in $G$, then $HK$ is normal in $G$ (product of normal subgroups is normal) This is a proof I couldn't find anywhere. Could somebody give me a help?
I need this to show that $$\frac{H}{H\cap K}\cong \frac{HK}{K}$$
but to form the quotient group I need first to show that $H\cap K$ is normal in $H$ and $K$ is normal in $HK$.

This is not hard to see :
$$
gHKg^{-1} = (gHg^{-1})(gKg^{-1}) \subseteq HK. 
$$
That is, any element of the form $ghkg^{-1}$ can be written in the form $(ghg^{-1})(gkg^{-1})$, the first factor being in $H$ and the second in $K$. 
Hope that helps,
