Group homomorphism/Kernel/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 of ${\displaystyle {}\varphi }$ is a subgroup

of ${\displaystyle {}G}$.