Editing Group action (section)

=== Right group action ===
Likewise, a ''right group action'' of <math>G</math> on <math>X</math> is a function
: <math>\alpha : X \times G \to X,</math>
that satisfies the analogous axioms:<ref>{{cite web |title=Definition:Right Group Action Axioms |url=https://proofwiki.org/wiki/Definition:Right_Group_Action_Axioms |website=Proof Wiki |access-date=19 December 2021}}</ref>
: {|
 |Identity:
 |<math>\alpha(x,e)=x</math>
|-
 |Compatibility:
 |<math>\alpha(\alpha(x,g),h)=\alpha(x,gh)</math>
|}
(with {{math|''&alpha;''(''x'', ''g'')}} often shortened to {{math|''xg''}} or {{math|''x''&sdot;''g''}} when the action being considered is clear from context)
: {|
 |Identity:
 |<math>x{\cdot}e = x</math>
|-
 |Compatibility:
 |<math>(x{\cdot}g){\cdot}h = x{\cdot}(gh)</math>
|}

for all {{mvar|g}} and {{mvar|h}} in {{mvar|G}} and all {{mvar|x}} in {{mvar|X}}.

The difference between left and right actions is in the order in which a product {{math|''gh''}} acts on {{mvar|x}}. For a left action, {{mvar|h}} acts first, followed by {{mvar|g}} second.  For a right action, {{mvar|g}} acts first, followed by {{mvar|h}} second. Because of the formula {{math|1=(''gh'')<sup>−1</sup> = ''h''<sup>−1</sup>''g''<sup>−1</sup>}}, a left action can be constructed from a right action by composing with the inverse operation of the group. Also, a right action of a group {{mvar|G}} on {{mvar|X}} can be considered as a left action of its [[opposite group]] {{math|''G''<sup>op</sup>}} on {{mvar|X}}.

Thus, for establishing general properties of group actions, it suffices to consider only left actions. However, there are cases where this is not possible. For example, the multiplication of a group [[Induced representation|induces]] both a left action and a right action on the group itself—multiplication on the left and on the right, respectively.