Editing Quotient group (section)

== Properties ==
The quotient group <math>G\,/\,G</math> is [[group isomorphism|isomorphic]] to the [[trivial group]] (the group with one element), and <math>G\,/\,\left\{e \right\}</math> is isomorphic to {{tmath|1= G }}.

The [[group order|order]] of {{tmath|1= G\,/\,N }}, by definition the number of elements, is equal to {{tmath|1= \vert G : N \vert }}, the [[index of a subgroup|index]] of <math>N</math> in {{tmath|1= G }}. If <math>G</math> is finite, the index is also equal to the order of <math>G</math> divided by the order of {{tmath|1= N }}. The set <math>G\,/\,N</math> may be finite, although both <math>G</math> and <math>N</math> are infinite (for example, {{tmath|1= \Z\,/\,2\Z }}).

There is a "natural" [[surjective]] [[group homomorphism]] {{tmath|1= \pi: G \rightarrow G\,/\,N }}, sending each element <math>g</math> of <math>G</math> to the coset of <math>N</math> to which <math>g</math> belongs, that is: {{tmath|1= \pi(g) = gN }}. The mapping <math>\pi</math> is sometimes called the ''canonical projection of <math>G</math> onto {{tmath|1= G\,/\,N }}''. Its [[kernel (algebra)|kernel]] is {{tmath|1= N }}.

There is a bijective correspondence between the subgroups of <math>G</math> that contain <math>N</math> and the subgroups of {{tmath|1= G\,/\,N }}; if <math>H</math> is a subgroup of <math>G</math> containing {{tmath|1= N }}, then the corresponding subgroup of <math>G\,/\,N</math> is {{tmath|1= \pi(H) }}. This correspondence holds for normal subgroups of <math>G</math> and <math>G\,/\,N</math> as well, and is formalized in the [[lattice theorem]].

Several important properties of quotient groups are recorded in the [[fundamental theorem on homomorphisms]] and the [[isomorphism theorem]]s.

If <math>G</math> is [[abelian group|abelian]], [[nilpotent group|nilpotent]], [[solvable group|solvable]], [[cyclic group|cyclic]] or [[generating set of a group|finitely generated]], then so is {{tmath|1= G\,/\,N }}.

If <math>H</math> is a subgroup in a finite group {{tmath|1= G }}, and the order of <math>H</math> is one half of the order of {{tmath|1= G }}, then <math>H</math> is guaranteed to be a normal subgroup, so <math>G\,/\,H</math> exists and is isomorphic to {{tmath|1= \mathrm{C}_2 }}.  This result can also be stated as "any subgroup of index 2 is normal", and in this form it applies also to infinite groups. Furthermore, if <math>p</math> is the smallest prime number dividing the order of a finite group, {{tmath|1= G }}, then if <math>G\,/\,H</math> has order {{tmath|1= p }}, <math>H</math> must be a normal subgroup of {{tmath|1= G }}.<ref>{{harvtxt|Dummit|Foote|2003|p=120}}</ref>

Given <math>G</math> and a normal subgroup {{tmath|1= N }}, then <math>G</math> is a [[group extension]] of <math>G\,/\,N</math> by {{tmath|1= N }}. One could ask whether this extension is trivial or split; in other words, one could ask whether <math>G</math> is a [[direct product of groups|direct product]] or [[semidirect product]] of <math>N</math> and {{tmath|1= G\,/\,N }}. This is a special case of the [[extension problem]]. An example where the extension is not split is as follows: Let {{tmath|1= G = \mathrm{Z}_4 = \left\{0, 1, 2, 3 \right\} }}, and {{tmath|1= N = \left\{0, 2 \right\} }}, which is isomorphic to {{tmath|1= \mathrm{Z}_2 }}. Then <math>G\,/\,N</math> is also isomorphic to {{tmath|1= \mathrm{Z}_2 }}. But <math>\mathrm{Z}_2</math> has only the trivial [[automorphism]], so the only semi-direct product of <math>N</math> and <math>G\,/\,N</math> is the direct product. Since <math>\mathrm{Z}_4</math> is different from {{tmath|1= \mathrm{Z}_2 \times \mathrm{Z}_2 }}, we conclude that <math>G</math> is not a semi-direct product of <math>N</math> and {{tmath|1= G\,/\,N }}.