Kernels are Normal Subgroups

Kernels are Normal Subgroups: Let $f:G\to H$ be a group homomorphism between groups. Then the kernel $\ker(f)$ is a normal subgroup of $G$.

This fact is what makes quotient groups naturally arise from homomorphisms, and it underlies the first isomorphism theorem .

Proof sketch: First, $\ker(f)$ is a subgroup by a subgroup test. For normality, if $k\in\ker(f)$ and $g\in G$, then $f(gkg^{-1})=f(g)f(k)f(g)^{-1}=f(g)e f(g)^{-1}=e$, so $gkg^{-1}\in\ker(f)$; apply the normality criterion.