Group homomorphism/Factorization/Fact

${\displaystyle G{\stackrel {q}{\longrightarrow }}G/\operatorname {kern} \varphi {\stackrel {\theta }{\longrightarrow }}\operatorname {Im} \varphi {\stackrel {\iota }{\hookrightarrow }}H,}$

where ${\displaystyle {}q}$ is the canonical projection, ${\displaystyle {}\theta }$ is a group isomorphism, and ${\displaystyle {}\iota }$ is the canonical inclusion of the