Editing Group action (section)

== Definition ==

=== Left group action ===
If <math>G</math> is a [[Group (mathematics)|group]] with [[identity element]] <math>e</math>, and <math>X</math> is a set, then a (''left'') ''group action'' <math>\alpha</math> of <math>G</math> on {{mvar|X}} is a [[Function (mathematics)|function]]
: <math>\alpha : G \times X \to X</math>
that satisfies the following two [[axioms]]:<ref>{{cite book|author=Eie & Chang |title=A Course on Abstract Algebra|year=2010|url={{Google books|plainurl=y|id=jozIZ0qrkk8C|page=144|text=group action}}|page=144}}</ref>
: {|
|Identity:
|<math>\alpha(e,x)=x</math>
|-
|Compatibility:
|<math>\alpha(g,\alpha(h,x))=\alpha(gh,x)</math>
|}
for all {{mvar|g}} and {{mvar|h}} in {{mvar|G}} and all {{mvar|x}} in <math>X</math>.

The group <math>G</math> is then said to act on <math>X</math> (from the left). A set <math>X</math> together with an action of <math>G</math> is called a (''left'') <math>G</math>-''set''.

It can be notationally convenient to [[currying|curry]] the action <math>\alpha</math>, so that, instead, one has a collection of [[transformation (geometry)|transformations]] {{math|''&alpha;''<sub>''g''</sub> : ''X'' → ''X''}}, with one transformation {{math|''&alpha;''<sub>''g''</sub>}} for each group element {{math|''g'' ∈ ''G''}}. The identity and compatibility relations then read
: <math>\alpha_e(x) = x</math>
and
: <math>\alpha_g(\alpha_h(x)) = (\alpha_g \circ \alpha_h)(x) = \alpha_{gh}(x)</math>
The second axiom states that the function composition is compatible with the group multiplication; they form a [[commutative diagram]]. This axiom can be shortened even further, and written as <math>\alpha_g\circ\alpha_h=\alpha_{gh}</math>.

With the above understanding, it is very common to avoid writing <math>\alpha</math> entirely, and to replace it with either a dot, or with nothing at all. Thus, {{math|''&alpha;''(''g'', ''x'')}} can be shortened to {{math|''g''&sdot;''x''}} or {{math|''gx''}}, especially when the action is clear from context. The axioms are then
: <math>e{\cdot}x = x</math>
: <math>g{\cdot}(h{\cdot}x) = (gh){\cdot}x</math>

From these two axioms, it follows that for any fixed {{mvar|g}} in <math>G</math>, the function from {{mvar|X}} to itself which maps {{mvar|x}} to {{math|''g''&sdot;''x''}} is a [[bijection]], with inverse bijection the corresponding map for {{math|''g''<sup>&minus;1</sup>}}. Therefore, one may equivalently define a group action of {{mvar|G}} on {{mvar|X}} as a group homomorphism from {{mvar|G}} into the symmetric group {{math|Sym(''X'')}} of all bijections from {{mvar|X}} to itself.<ref>This is done, for example, by {{cite book|author=Smith |title=Introduction to abstract algebra|year=2008|url={{Google books|plainurl=y|id=PQUAQh04lrUC|page=253|text=group action}}|page=253}}</ref>

=== 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.