Kernel is normal

Proposition (Kernel is normal). Let $f:G\to H$ be a group homomorphism . Let $\ker(f)$ be its kernel . Then $\ker(f)$ is a normal subgroup of $G$.

Context. Normal subgroups arise naturally as kernels; conversely, every normal subgroup is the kernel of a canonical map to a quotient group.

Proof sketch. Let $k\in \ker(f)$ and $g\in G$. Then

so $gkg^{-1}\in \ker(f)$. Hence $\ker(f)\lhd G$.