Group/Normal subgroup/Residue class group/Fact

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

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

denote the canonical projection.

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