Editing Group action (section)

== Examples ==
* The ''{{visible anchor|trivial}}'' action of any group {{math|''G''}} on any set {{math|''X''}} is defined by {{math|1=''g''⋅''x'' = ''x''}} for all {{math|''g''}} in {{math|''G''}} and all {{math|''x''}} in {{math|''X''}}; that is, every group element induces the [[identity function|identity permutation]] on {{math|''X''}}.<ref>{{cite book|author=Eie & Chang |title=A Course on Abstract Algebra|year=2010|url={{Google books|plainurl=y|id=jozIZ0qrkk8C|page=144|text=trivial action}}|page=145}}</ref>
* In every group {{math|''G''}}, left multiplication is an action of {{math|''G''}} on {{math|''G''}}: {{math|1=''g''⋅''x'' = ''gx''}} for all {{math|''g''}}, {{math|''x''}} in {{math|''G''}}. This action is free and transitive (regular), and forms the basis of a rapid proof of [[Cayley's theorem]] – that every group is isomorphic to a subgroup of the symmetric group of permutations of the set {{math|''G''}}.
* In every group {{math|''G''}} with subgroup {{math|''H''}}, left multiplication is an action of {{math|''G''}} on the set of cosets {{math|''G'' / ''H''}}: {{math|1=''g''⋅''aH'' = ''gaH''}} for all {{math|''g''}}, {{math|''a''}} in {{math|''G''}}. In particular if {{math|''H''}} contains no nontrivial [[normal subgroups]] of {{math|''G''}} this induces an isomorphism from {{math|''G''}} to a subgroup of the permutation group of [[Degree of a permutation group|degree]] {{math|[''G'' : ''H'']}}.
* In every group {{math|''G''}}, [[inner automorphism|conjugation]] is an action of {{math|''G''}} on {{math|''G''}}: {{math|1=''g''⋅''x'' = ''gxg''<sup>−1</sup>}}. An exponential notation is commonly used for the right-action variant: {{math|1=''x<sup>g</sup>'' = ''g''<sup>−1</sup>''xg''}}; it satisfies ({{math|1=''x''<sup>''g''</sup>)<sup>''h''</sup> = ''x''<sup>''gh''</sup>}}.
* In every group {{math|''G''}} with subgroup {{math|''H''}}, conjugation is an action of {{math|''G''}} on conjugates of {{math|''H''}}: {{math|1=''g''⋅''K'' = ''gKg''<sup>−1</sup>}} for all {{math|''g''}} in {{math|''G''}} and {{math|''K''}} conjugates of {{math|''H''}}.
* An action of {{math|'''Z'''}} on a set {{math|''X''}} uniquely determines and is determined by an [[automorphism]] of {{math|''X''}}, given by the action of 1. Similarly, an action of {{math|'''Z''' / 2'''Z'''}} on {{math|''X''}} is equivalent to the data of an [[involution (mathematics)|involution]] of {{math|''X''}}.
* The symmetric group {{math|S<sub>''n''</sub>}} and its subgroups act on the set {{math|{{mset|1, ..., ''n''}}}} by permuting its elements
* The [[symmetry group]] of a polyhedron acts on the set of vertices of that polyhedron. It also acts on the set of faces or the set of edges of the polyhedron.
* The symmetry group of any geometrical object acts on the set of points of that object.
* For a [[coordinate space]] {{math|''V''}} over a field {{math|''F''}} with group of units {{math|''F''*}}, the mapping {{math|''F''* × ''V'' → ''V''}} given by {{math|''a'' × (''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>) ↦ (''ax''<sub>1</sub>, ''ax''<sub>2</sub>, ..., ''ax''<sub>''n''</sub>)}} is a group action called [[scalar multiplication]].
* The automorphism group of a vector space (or [[graph theory|graph]], or group, or ring&nbsp;...) acts on the vector space (or set of vertices of the graph, or group, or ring ...).
* The general linear group {{math|GL(''n'', ''K'')}} and its subgroups, particularly its [[Lie subgroup]]s (including the special linear group {{math|SL(''n'', ''K'')}}, [[orthogonal group]] {{math|O(''n'', ''K'')}}, special orthogonal group {{math|SO(''n'', ''K'')}}, and [[symplectic group]] {{math|Sp(''n'', ''K'')}}) are [[Lie group]]s that act on the vector space {{math|''K''<sup>''n''</sup>}}. The group operations are given by multiplying the matrices from the groups with the vectors from {{math|''K''<sup>''n''</sup>}}.
* The general linear group {{math|GL(''n'', '''Z''')}} acts on {{math|'''Z'''<sup>''n''</sup>}} by natural matrix action. The orbits of its action are classified by the [[greatest common divisor]] of coordinates of the vector in {{math|'''Z'''<sup>''n''</sup>}}.
* The [[affine group]] acts [[#Notable properties of actions|transitively]] on the points of an [[affine space]], and the subgroup V of the affine group (that is, a vector space) has transitive and free (that is, ''regular'') action on these points;<ref>{{cite book|title=Geometry and topology|last=Reid|first=Miles|publisher=Cambridge University Press|year=2005|isbn=9780521613255|location=Cambridge, UK New York|pages=170}}</ref> indeed this can be used to give a definition of an [[Affine space#Definition|affine space]].
* The [[projective linear group]] {{math|PGL(''n'' + 1, ''K'')}} and its subgroups, particularly its Lie subgroups, which are Lie groups that act on the [[projective space]] {{math|'''P'''<sup>n</sup>(''K'')}}. This is a quotient of the action of the general linear group on projective space. Particularly notable is {{math|PGL(2, ''K'')}}, the symmetries of the projective line, which is sharply 3-transitive, preserving the [[cross ratio]]; the [[Möbius group]] {{math|PGL(2, '''C''')}} is of particular interest.
* The [[Isometry|isometries]] of the plane act on the set of 2D images and patterns, such as [[wallpaper group|wallpaper pattern]]s. The definition can be made more precise by specifying what is meant by image or pattern, for example, a function of position with values in a set of colors. Isometries are in fact one example of affine group (action).{{dubious|reason=The isometries of a space are a subgroup of the affine group of that space, but not an affine group in themselves|date=March 2015}}
* The sets acted on by a group {{math|''G''}} comprise the [[Category (mathematics)|category]] of {{math|''G''}}-sets in which the objects are {{math|''G''}}-sets and the [[morphism]]s are {{math|''G''}}-set homomorphisms: functions {{math|''f'' : ''X'' → ''Y''}} such that {{math|1=''g''⋅(''f''(''x'')) = ''f''(''g''⋅''x'')}} for every {{math|''g''}} in {{math|''G''}}.
* The [[Galois group]] of a [[field extension]] {{math|''L'' / ''K''}} acts on the field {{math|''L''}} but has only a trivial action on elements of the subfield {{math|''K''}}. Subgroups of {{math|Gal(''L'' / ''K'')}} correspond to subfields of {{math|''L''}} that contain {{math|''K''}}, that is, intermediate field extensions between {{math|''L''}} and {{math|''K''}}.
* The additive group of the [[real number]]s {{math|('''R''', +)}} acts on the [[phase space]] of "[[well-behaved]]" systems in [[classical mechanics]] (and in more general [[dynamical systems]]) by [[time translation]]: if {{math|''t''}} is in {{math|'''R'''}} and {{math|''x''}} is in the phase space, then {{math|''x''}} describes a state of the system, and {{math|''t'' + ''x''}} is defined to be the state of the system {{math|''t''}} seconds later if {{math|''t''}} is positive or {{math|&minus;''t''}} seconds ago if {{math|''t''}} is negative.
*The additive group of the real numbers {{math|('''R''', +)}} acts on the set of real [[Function of a real variable|functions of a real variable]] in various ways, with {{math|(''t''⋅''f'')(''x'')}} equal to, for example, {{math|''f''(''x'' + ''t'')}}, {{math|''f''(''x'') + ''t''}}, {{math|''f''(''xe<sup>t</sup>'')}}, {{math|''f''(''x'')''e''<sup>''t''</sup>}}, {{math|''f''(''x'' + ''t'')''e<sup>t</sup>''}}, or {{math|''f''(''xe''<sup>''t''</sup>) + ''t''}}, but not {{math|''f''(''xe<sup>t</sup>'' + ''t'')}}.
* Given a group action of {{math|''G''}} on {{math|''X''}}, we can define an induced action of {{math|''G''}} on the [[power set]] of {{math|''X''}}, by setting {{math|1=''g''⋅''U'' = {''g''⋅''u'' : ''u'' ∈ ''U''}<nowiki/>}} for every subset {{math|''U''}} of {{math|''X''}} and every {{math|''g''}} in {{math|''G''}}. This is useful, for instance, in studying the action of the large [[Mathieu group]] on a 24-set and in studying symmetry in certain models of [[finite geometry|finite geometries]].
* The [[quaternion]]s with [[Norm of a quaternion|norm]] 1 (the [[versor]]s), as a multiplicative group, act on {{math|'''R'''<sup>3</sup>}}: for any such quaternion {{math|1=''z'' = cos ''α''/2 + '''v''' sin ''α''/2}}, the mapping {{math|1=''f''('''x''') = ''z'''''x'''''z''<sup>*</sup>}} is a counterclockwise rotation through an angle {{math|''α''}} about an axis given by a unit vector {{math|'''v'''}}; {{math|''z''}} is the same rotation; see [[quaternions and spatial rotation]]. This is not a faithful action because the quaternion {{math|−1}} leaves all points where they were, as does the quaternion {{math|1}}.
* Given left {{math|''G''}}-sets {{math|''X''}}, {{math|''Y''}}, there is a left {{math|''G''}}-set {{math|''Y''{{i sup|''X''}}}} whose elements are {{math|''G''}}-equivariant maps {{math|''&alpha;'' : ''X'' × ''G'' → ''Y''}}, and with left {{math|''G''}}-action given by {{math|1=''g''⋅''&alpha;'' = ''&alpha;'' ∘ (id<sub>''X''</sub> × –''g'')}} (where "{{math|–''g''}}" indicates right multiplication by {{math|''g''}}). This {{math|''G''}}-set has the property that its fixed points correspond to equivariant maps {{math|''X'' → ''Y''}}; more generally, it is an [[exponential object]] in the category of {{math|''G''}}-sets.