Group homomorphism/Surjective and residue class group/Fact

Isomorphism theorem (groups)

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

$\varphi \colon G\longrightarrow H$

be a surjective group homomorphism.

Then there exists a canonical

isomorphism

${\tilde {\varphi }}\colon G/\operatorname {kern} \varphi \longrightarrow H.$

Proof, Write another proof