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==Definition==
Let {{M|(G,\times)}} be a [[group]]. Let {{M|H\subseteq G}} be a [[normal subgroup]] of {{M|G}}. Then:
* the {{link|coset|group|s}} of {{M|H}} in {{M|G}} form a group whose operation is the group [[operation on subsets]]<ref group="Note">Let {{M|A\subseteq G}} be an arbitrary subset of a [[group]] {{M|(G,\times)}} and let {{M|g\in G}} be given. Then:
* {{M|1=g\times A:=\{g\times a\ \vert\ a\in A\} }}
By itself this isn't very interesting.</ref>
We denote this new group {{M|\frac{G}{H} }} or {{M|G/H}}.


With it we also get a [[group homomorphism]] called the [[canonical projection of the quotient group]]:
* {{M|\pi:G\rightarrow\frac{G}{H} }} given by {{M|\pi:g\rightarrow [g]}} where {{M|[g]}} denotes the {{link|coset|group}} containing {{M|g}}.
{{Caution|This requires some work, why must the cosets of {{M|H}} in {{M|G}} partition {{M|G}}?}}
==See also==
* [[Normal subgroup]]
** [[Every subgroup of an abelian group is normal]]
==Notes==
<references group="Note"/>
==References==
<references/>
{{Definition|Abstract Algebra|Group Theory}}