Group/Normal subgroup/Residue class group/Fact

Let ${}G$ be a group, and let ${}H\subseteq G$ be a normal subgroup. Let ${}G/H$ be the set of all cosets (the quotient set), and let

$q\colon G\longrightarrow G/H,g\longmapsto [g],$

denote the canonical projection.

Then there exists a uniquely determined group structure on ${}G/H$ such that ${}q$ is a group homomorphism.

Proof, Write another proof