Editing Coset (section)

=== 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''}}