Normal subgroup/Residue class group/Introduction/Section

Theorem

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

Since the canonical projection shall be a group homomorphism, the operation must fulfill

${}[x][y]=[xy]\,.$

We have to show that this rule gives a well-defined operation on ${}G/H$, that is, is independent of the choice of representatives. Hence, we have to show for ${}[x]=[x']$ and ${}[y]=[y']$ that ${}[xy]=[x'y']$ holds. Due to the condition, we can write ${}x'=xh$ and ${}hy'={\tilde {h}}y=yh'$ with ${}h,{\tilde {h}},h'\in H$. Therefore,

${}x'y'=(xh)y'=x(hy')=x(yh')=xyh'\,.$

This means ${}[xy]=[x'y']$. From this, the group property, the homomorphism property of the projection and the uniqueness follows.

$\Box$

Definition

Let ${}G$ be a group, and let ${}H\subseteq G$ be a normal subgroup. The quotient set

$G/H,$

endowed with the (according to fact) uniquely determined group structure, is called the factor group of ${}G$ modulo ${}H$. The elements ${}[g]\in G/H$ are called residue classes. For a residue class ${}[g]$, every element ${}g'\in G$ with ${}[g']=[g]$

is called a representative of ${}[g]$.