The kernel is a normal subgroup

Statement

Let $G$ and $H$ be groups and let $\varphi \colon G \to H$ be a group homomorphism. Then the kernel $\ker \varphi$ is a normal subgroup of $G$.

Remarks

For $g \in G$ and $k \in \ker \varphi$, $\varphi(gkg^{-1}) = \varphi(g) \, e_H \, \varphi(g)^{-1} = e_H$. Conversely, each normal subgroup $N$ of $G$ is the kernel of the projection $G \to G/N$. Thus the normal subgroups of $G$ are exactly the kernels of the homomorphisms from $G$.

Explanations

L1 human

If an element goes to the identity, then each of its conjugates also goes to the identity. So the kernel is closed under conjugation, which is the definition of a normal subgroup. This is one reason why normal subgroups are important.