Editing Coset (section)

== As orbits of a group action ==
{{main|Group action}}

A subgroup {{mvar|H}} of a group {{mvar|G}} can be used to define an [[group action|action]] of {{mvar|H}} on {{mvar|G}} in two natural ways. A ''right action'', {{math|''G'' × ''H'' → ''G''}} given by {{math|(''g'', ''h'') → ''gh''}} or a ''left action'', {{math|''H'' × ''G'' → ''G''}} given by {{math|(''h'', ''g'') → ''hg''}}. The [[Orbit (group theory)|orbit]] of {{mvar|g}} under the right action is the left coset {{mvar|gH}}, while the orbit under the left action is the right coset {{mvar|Hg}}.<ref name=Jacobson>{{harvnb|Jacobson|2009|loc=p. 52}}</ref>