Group homomorphism/Factorization/Fact

Factorization theorem (groups)

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

$\varphi \colon G\longrightarrow H$

be a group homomorphism.

Then there exists a canonical factorization

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

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

image group.

Proof, Write another proof