Editing Group (mathematics) (section)

=== Quotient groups ===
{{Main|Quotient group}}
Suppose that <math>N</math> is a normal subgroup of a group {{tmath|1= G }}, and
<math display=block>G/N = \{gN \mid g\in G\}</math>
denotes its set of cosets.
Then there is a unique group law on <math>G/N</math> for which the map <math>G\to G/N</math> sending each element <math>g</math> to <math>gN</math> is a homomorphism.
Explicitly, the product of two cosets <math>gN</math> and <math>hN</math> is {{tmath|1= (gh)N }}, the coset <math>eN = N</math> serves as the identity of {{tmath|1= G/N }}, and the inverse of <math>gN</math> in the quotient group is {{tmath|1= (gN)^{-1} = \left(g^{-1}\right)N }}.
The group {{tmath|1= G/N }}, read as "{{tmath|1= G }} modulo {{tmath|1= N }}",{{sfn|Lang|2005|loc=§II.4|p=45}} is called a ''quotient group'' or ''factor group''.
The quotient group can alternatively be characterized by a [[universal property]].

{| class="wikitable" style="float:right; text-align:center; margin:.5em 0 .5em 1em; width:200px;"
|+ Cayley table of the quotient group <math>\mathrm{D}_4/R</math>
|-
! style="width:30px;"| <math>\cdot</math>
! style="width:33%;"| <math>R</math>
! style="width:33%;"| <math>U</math>
|-
! <math>R</math>
| <math>R</math> || <math>U</math>
|-
! <math>U</math>
| <math>U</math> || <math>R</math>
|}
The elements of the quotient group <math>\mathrm{D}_4/R</math> are <math>R</math> and {{tmath|1= U=f_{\mathrm{v} }R }}. The group operation on the quotient is shown in the table. For example, {{tmath|1= U\cdot U=f_{\mathrm{v} }R\cdot f_{\mathrm{v} }R=(f_{\mathrm{v} }\cdot f_{\mathrm{v} })R=R }}. Both the subgroup <math>R=\{\mathrm{id},r_1,r_2,r_3\}</math> and the quotient <math>\mathrm{D}_4/R</math> are abelian, but <math>\mathrm{D}_4</math> is not. Sometimes a group can be reconstructed from a subgroup and quotient (plus some additional data), by the [[semidirect product]] construction; <math>\mathrm{D}_4</math> is an example.

The [[first isomorphism theorem]] implies that any [[surjective]] homomorphism <math>\phi : G \to H</math> factors canonically as a quotient homomorphism followed by an isomorphism: {{tmath|1= G \to G/\ker \phi \;\stackrel{\sim}{\to}\; H }}.
Surjective homomorphisms are the [[epimorphism]]s in the category of groups.