If $H$ is normal in $G$ then $H$ is the kernel of a group homomorphism.

Question

Let $G$ a group and $H$ a subgroup of $G$. Show that if $H$ is normal, then it's the kernel of a group homomorphism.

Attempt

I proved that if $H$ is the kernel of a group homomorphism, then $H$ is normal. But how can I prove the converse ? I tried to construct something as $f:G\to G$ s.t. $f(gh)=g$. Then $\ker f=H$, but I have difficulty to show that $f$ is indeed a homomorphism.

Answer 1

$H$ is the kernel of the canonical surjection $G \to G/H$ where $G/H$ is the quotient group.