Editing Coset (section)

== Definition ==
Let {{mvar|H}} be a subgroup of the group {{mvar|G}} whose operation is written multiplicatively (juxtaposition denotes the group operation). Given an element {{mvar|g}} of {{mvar|G}}, the '''left cosets''' of {{mvar|H}} in {{mvar|G}} are the sets obtained by multiplying each element of {{mvar|H}} by a fixed element {{mvar|g}} of {{mvar|G}} (where {{mvar|g}} is the left factor). In symbols these are,
{{block indent|em=1.5|text={{math|1=''gH'' = {{mset|''gh'' : ''h'' an element of ''H''}}}} for {{mvar|g}} in {{mvar|G}}.}}
The '''right cosets''' are defined similarly, except that the element {{mvar|g}} is now a right factor, that is,
{{block indent|em=1.5|text={{math|1=''Hg'' = {{mset|''hg'' : ''h'' an element of ''H''}}}} for {{mvar|g}} in {{mvar|G}}.}}

As {{mvar|g}} varies through the group, it would appear that many cosets (right or left) would be generated. Nevertheless, it turns out that any two left cosets (respectively right cosets) are either disjoint or are identical as sets.<ref name=Rotman2006>{{harvnb|Rotman|2006|loc=p. 156}}</ref>

If the group operation is written additively, as is often the case when the group is [[abelian group|abelian]], the notation used changes to {{math|''g'' + ''H''}} or {{math|''H'' + ''g''}}, respectively.

The symbol ''G''/''H'' is sometimes used for the set of (left) cosets {''gH'' : ''g'' an element of ''G''} (see below for a extension to right cosets and double cosets).  However, some authors (including Dummit & Foote and Rotman) reserve this notation specifically for representing the [[quotient group]] formed from the cosets in the case where ''H'' is a ''normal'' subgroup of ''G.'' 

=== First example ===
Let {{mvar|G}} be the [[dihedral group of order 6|dihedral group of order six]]. Its elements may be represented by {{math|{{mset|''I'', ''a'', ''a''<sup>2</sup>, ''b'', ''ab'', ''a''<sup>2</sup>''b''}}}}. In this group, {{math|1=''a''<sup>3</sup> = ''b''<sup>2</sup> = ''I''}} and {{math|1=''ba'' = ''a''<sup>2</sup>''b''}}. This is enough information to fill in the entire [[Cayley table]]:
{| class="wikitable" style="text-align:center"
!∗||{{mvar|I}} ||{{mvar|a}} ||{{math|''a''<sup>2</sup>}} ||{{mvar|b}} ||{{mvar|ab}} ||{{math|''a''<sup>2</sup>''b''}}
|-
!{{mvar|I}}
||{{mvar|I}} ||{{mvar|a}} ||{{math|''a''<sup>2</sup>}} ||{{mvar|b}} ||{{mvar|ab}} ||{{math|''a''<sup>2</sup>''b''}}
|-
!{{mvar|a}}
|{{mvar|a}} ||{{math|''a''<sup>2</sup>}} ||{{mvar|I}} ||{{mvar|ab}} ||{{math|''a''<sup>2</sup>''b''}} ||{{mvar|b}}
|-
!{{math|''a''<sup>2</sup>}}
|{{math|''a''<sup>2</sup>}} ||{{mvar|I}} ||{{mvar|a}} ||{{math|''a''<sup>2</sup>''b''}} ||{{mvar|b}} ||{{mvar|ab}}
|-
!{{mvar|b}}
|{{mvar|b}} ||{{math|''a''<sup>2</sup>''b''}} ||{{mvar|ab}} ||{{mvar|I}} ||{{math|''a''<sup>2</sup>}} || {{mvar| a}}
|-
!{{mvar|ab}}
|{{mvar|ab}} ||{{mvar|b}} ||{{math|''a''<sup>2</sup>''b''}} ||{{mvar|a}} || {{mvar|I}} ||{{math|''a''<sup>2</sup>}}
|-
!{{math|''a''<sup>2</sup>''b''}}
|{{math|''a''<sup>2</sup>''b''}} ||{{mvar|ab}} ||{{mvar|b}} || {{math|''a''<sup>2</sup>}} ||{{mvar|a}} || {{mvar|I}}
|}

Let {{mvar|T}} be the subgroup {{math|{{mset|''I'', ''b''}}}}. The (distinct) left cosets of {{mvar|T}} are:
* {{math|1=''IT'' = ''T'' = {{mset|''I'', ''b''}}}},
* {{math|1=''aT'' = {{mset|''a'', ''ab''}}}}, and
* {{math|1=''a''<sup>2</sup>''T'' = {{mset|''a''<sup>2</sup>, ''a''<sup>2</sup>''b''}}}}.
Since all the elements of {{mvar|G}} have now appeared in one of these cosets, generating any more can not give new cosets; any new coset would have to have an element in common with one of these and therefore would be identical to one of these cosets. For instance, {{math|1=''abT'' = {{mset|''ab'', ''a''}} = ''aT''}}.

The right cosets of {{mvar|T}} are:
* {{math|1=''TI'' = ''T'' = {{mset|''I'', ''b''}}}},
* {{math|1=''Ta'' = {{mset|''a'', ''ba''}} = {{mset|''a'', ''a''<sup>2</sup>''b''}}}} , and
* {{math|1=''Ta''<sup>2</sup> = {{mset|''a''<sup>2</sup>, ''ba''<sup>2</sup>}} = {{mset|''a''<sup>2</sup>, ''ab''}}}}.

In this example, except for {{mvar|T}}, no left coset is also a right coset.

Let {{mvar|H}} be the subgroup {{math|{{mset|''I'', ''a'', ''a''<sup>2</sup>}}}}. The left cosets of {{mvar|H}} are {{math|1=''IH'' = ''H''}} and {{math|1=''bH'' = {{mset|''b'', ''ba'', ''ba''<sup>2</sup>}}}}. The right cosets of {{mvar|H}} are {{math|1=''HI'' = ''H''}} and {{math|1=''Hb'' = {{mset|''b'', ''ab'', ''a''<sup>2</sup>''b''}} = {{mset|''b'', ''ba''<sup>2</sup>, ''ba''}}}}. In this case, every left coset of {{mvar|H}} is also a right coset of {{mvar|H}}.<ref name=Dean>{{harvnb|Dean|1990|loc=p. 100}}</ref>

Let {{mvar|H}} be a subgroup of a group {{mvar|G}} and suppose that {{math|''g''<sub>1</sub>}}, {{math|''g''<sub>2</sub> ∈ ''G''}}. The following statements are equivalent:<ref>{{Cite web|url=http://abstract.ups.edu/aata/section-cosets.html|title=AATA Cosets|access-date=2020-12-09|archive-date=2022-01-22|archive-url=https://web.archive.org/web/20220122151749/http://abstract.ups.edu/aata/section-cosets.html|url-status=dead}}</ref>
* {{math|1=''g''<sub>1</sub>''H'' = ''g''<sub>2</sub>''H''}}
* {{math|1=''Hg''<sub>1</sub><sup>−1</sup> = ''Hg''<sub>2</sub><sup>−1</sup>}}
* {{math|1=''g''<sub>1</sub>''H'' ⊂ ''g''<sub>2</sub>''H''}}
* {{math|1=''g''<sub>2</sub> ∈ ''g''<sub>1</sub>''H''}}
* {{math|1=''g''<sub>1</sub><sup>−1</sup>''g''<sub>2</sub> ∈ ''H''}}