A homomorphism's kernel is a normal subgroup

up:: Normal Subgroup

Given a Group Homomorphism $\phi :G\to G^{\prime }$, then this homomorphism’s Kernel (Group) is a normal subgroup of $G$.

Let $K:=\ker \phi$. Then we seek to prove

Let $g\in G$ and $k\in K$. Then

$$\phi (g^{-1}ng)=\phi (g{)}^{-1}\phi (k)\phi (g)=\phi (g{)}^{-1}\phi (g)=e_{G^{\prime }}$$

Thus, $g^{-1}ng\in K$.

SInce it holds that

$$\forall g\in G,\forall k\in K:g^{-1}kg\in K$$

then $K$ is a normal subgroup of the domain of $\phi$.