Group homomorphism/Kernel/Normal subgroup/Fact

Let ${\displaystyle {}G}$ and ${\displaystyle {}H}$ be groups, and let

${\displaystyle \varphi \colon G\longrightarrow H}$

be a group homomorphism.

Then the

kernel ${\displaystyle {}\operatorname {kern} \varphi }$ is a normal subgroup

in ${\displaystyle {}G}$.