Quotient Group

The group of cosets of a normal subgroup

Quotient Group

Let $N$ be a normal subgroup of a group $G$ . The quotient group $G/N$ is the set of (left) cosets $\{gN : g\in G\}$ equipped with the operation

(gN)(hN) := (gh)N.

Normality of $N$ is exactly what makes this operation well-defined, meaning it does not depend on the choice of representatives $g,h$ of the cosets.

There is a canonical group homomorphism (the projection) $\pi:G\to G/N$ , $\pi(g)=gN$ , whose kernel is $N$ . Quotient groups are the objects that appear in exact sequences and in the first isomorphism theorem , which identifies $G/\ker(\varphi)$ with $\operatorname{im}(\varphi)$ for any homomorphism $\varphi$ .

Examples:

$\mathbb{Z}/n\mathbb{Z}$ is the quotient of $\mathbb{Z}$ by the subgroup $n\mathbb{Z}$ ; its elements are the residue classes mod $n$ .
In $S_3$ , the alternating subgroup $A_3$ is normal, and $S_3/A_3$ has order $2$ (isomorphic to $C_2$ ).
If $G$ is abelian, every subgroup is normal, so quotients $G/H$ exist for all subgroups $H\le G$ .