Group homomorphism/Homomorphism theorem/Surjective and kernel/Fact

Homomorphism theorem (groups)

Let ${}G,Q$ and ${}H$ be groups, and let ${}\varphi \colon G\rightarrow H$ be a group homomorphism and ${}\psi \colon G\rightarrow Q$ a surjective group homomorphism. Suppose that

${}\operatorname {kern} \psi \subseteq \operatorname {kern} \varphi \,.$

ist.

Then there exists a uniquely determined group homomorphism

${\tilde {\varphi }}\colon Q\longrightarrow H$

such that ${}\varphi ={\tilde {\varphi }}\circ \psi$

holds.

Put differently, the diagram

${\begin{matrix}G&{\stackrel {\varphi }{\longrightarrow }}&H&\\\!\!\!\!\!\psi \downarrow &\nearrow {\tilde {\varphi }}\!\!\!\!\!&\\Q&&&&\!\!\!\!\!\!\!\!\\\end{matrix}}$

commutes.

Proof, Write another proof