Group homomorphism/Factorization/Fact

Factorization theorem (groups)

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

$\phi :G⟶H$

be a group homomorphism.

Then there exists a canonical factorization

$G\stackrel{q}{⟶}G/\mathrm{kern}\phi \stackrel{\theta }{⟶}\mathrm{Im}\phi \stackrel{\iota }{\hookrightarrow }H,$

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