Group action

{{Short description|Transformations induced by a mathematical group}} {{About|the mathematical concept|the sociology term|group action (sociology)}} {{Group theory sidebar}} [[File:Group action on equilateral triangle.svg|right|thumb|The [[cyclic group]] {{math|C3}} consisting of the [[Rotation (mathematics)|rotations]] by 0°, 120° and 240° acts on the set of the three vertices.]]

In mathematics, an '''action''' of a group <math>G</math> on a [[set (mathematics)|set]] <math>S</math> is, loosely speaking, an operation that takes an element of <math>G</math> and an element of <math>S</math> and produces another element of <math>S.</math> More formally, it is a [[group homomorphism]] from <math>G</math> to the [[automorphism group]] of <math>S</math> (the set of all [[bijection]]s on <math>S</math> along with group operation being [[function composition]]). One says that <math>G</math> '''acts''' on <math>S.</math>

Many sets of [[transformation (function)|transformation]]s form a [[group (mathematics)|group]] under function composition; for example, the [[rotation (mathematics)|rotation]]s around a point in the plane. It is often useful to consider the group as an [[abstract group]], and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a [[mathematical structure|structure]] acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles.

If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of [[Euclidean isometry|Euclidean isometries]] acts on [[Euclidean space]] and also on the figures drawn in it; in particular, it acts on the set of all [[triangle]]s. Similarly, the group of [[symmetries]] of a [[polyhedron]] acts on the [[vertex (geometry)|vertices]], the [[edge (geometry)|edges]], and the [[face (geometry)|faces]] of the polyhedron.

A group action on a [[vector space]] is called a [[Group representation|representation]] of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with [[subgroups]] of the [[general linear group]] <math>\operatorname{GL}(n,K)</math>, the group of the [[invertible matrices]] of [[dimension]] <math>n</math> over a [[Field (mathematics)|field]] <math>K</math>.

The [[symmetric group]] <math>S_n</math> acts on any [[set (mathematics)|set]] with <math>n</math> elements by permuting the elements of the set. Although the group of all [[permutation]]s of a set depends formally on the set, the concept of group action allows one to consider a single group for studying the permutations of all sets with the same [[cardinality]].

== 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 <math>X</math> 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 <math>g</math> and <math>h</math> in <math>G</math> and all <math>x</math> 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_g:X\rightarrow X</math>, with one transformation <math>\alpha_g</math> for each group element <math>g\in G</math>. The identity and compatibility relations then read <math display="block">\alpha_e(x) = x</math> and <math display="block">\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)</math> can be shortened to <math>g\cdot x</math> or <math>gx</math>, especially when the action is clear from context. The axioms are then <math display="block">\left\{\begin{align}& e\cdot x=x \\ & g\cdot(h\cdot x)=(gh)\cdot x\end{align}\right.</math>

From these two axioms, it follows that for any fixed <math>g</math> in <math>G</math>, the function from <math>X</math> to itself which maps <math>x</math> to <math>g\cdot x</math> is a [[bijection]], with inverse bijection the corresponding map for <math>g^{-1}</math>. Therefore, one may equivalently define a group action of <math>G</math> on <math>X</math> as a group homomorphism from <math>G</math> into the [[symmetric group]] <math>\operatorname{Sym}(X)</math> of all bijections from <math>X</math> 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|''α''(''x'', ''g'')}} often shortened to {{math|''xg''}} or {{math|''x''⋅''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'')−1 = ''h''−1''g''−1}}, 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''op}} on {{mvar|X}}. Thus, for establishing general properties of a single group action, it suffices to consider only left actions.

== Notable properties of actions == Let <math>G</math> be a group acting on a set <math>X</math>. The action is called ''{{visible anchor|faithful}}'' or ''{{visible anchor|effective}}'' if <math>g\cdot x=x</math> for all <math>x\in X</math> implies that <math>g=e_G</math>. Equivalently, the [[homomorphism]] from <math>G</math> to the group of bijections of <math>X</math> corresponding to the action is [[injective]].

The action is called ''{{visible anchor|free}}'' (or ''semiregular'' or ''fixed-point free'') if the statement that <math>g\cdot x=x</math> for some <math>x\in X</math> already implies that <math>g=e_G</math>. In other words, no non-trivial element of <math>G</math> fixes a point of <math>X</math>. This is a much stronger property than faithfulness.

For example, the action of any group on itself by left multiplication is free. This observation implies [[Cayley's theorem]] that any group can be [[Embedding|embedded]] in a symmetric group (which is infinite when the group is). A finite group may act faithfully on a set of size much smaller than its cardinality (however such an action cannot be free). For instance the abelian 2-group <math>(\mathbb{Z}/2\mathbb{Z})^n</math> (of cardinality <math>2^n</math>) acts faithfully on a set of size <math>2n</math>. This is not always the case, for example the [[cyclic group]] <math>\mathbb{Z}/2^n\mathbb{Z}</math> cannot act faithfully on a set of size less than <math>2^n</math>.

In general, the smallest set on which a faithful action can be defined can vary greatly for groups of the same size. For example, three groups of size 120 are the symmetric group <math>S_5</math>, the [[icosahedral group]] <math>A_5\times\mathbb{Z}/2\mathbb{Z}</math> and the cyclic group <math>\mathbb{Z}/120\mathbb{Z}</math>. The smallest sets on which faithful actions can be defined for these groups are of size 5, 7, and 16 respectively.

=== Transitivity properties === The action of <math>G</math> on <math>X</math> is called ''{{visible anchor|transitive}}'' if for any two points <math>x,y\in X</math> there exists a <math>g\in G</math> so that <math>g\cdot x=y</math>.

The action is ''{{visible anchor|simply transitive}}'' (or ''sharply transitive'', or ''{{visible anchor|regular}}'') if it is both transitive and free. This means that given <math>x,y\in X</math> there is exactly one <math>g\in G</math> such that <math>g\cdot x=y</math>. If <math>X</math> is acted upon simply transitively by a group <math>G</math> then it is called a [[principal homogeneous space]] for <math>G</math> or a <math>G</math>-torsor.

For an integer <math>n\geq 1</math>, the action is {{visible anchor|n-transitive|text=''<math>n</math>-transitive''}} if <math>X</math> has at least <math>n</math> elements, and for any pair of <math>n</math>-tuples <math>(x_1,\ldots,x_n),(y_1,\ldots,y_n)\in X^n</math> with pairwise distinct entries (that is <math>x_i\neq x_j</math>, <math>y_i\neq y_j</math> when <math>i\neq j</math>) there exists a <math>g\in G</math> such that <math>g\cdot x_i=y_i</math> for <math>i=1,\ldots,n</math>. In other words, the action on the subset of <math>X^n</math> of tuples without repeated entries is transitive. For <math>n=2,3</math> this is often called double, respectively triple, transitivity. The class of [[2-transitive group]]s (that is, subgroups of a finite symmetric group whose action is 2-transitive) and more generally [[multiply transitive group]]s is well-studied in finite group theory.

An action is {{visible anchor|sharply n-transitive|text=''sharply <math>n</math>-transitive''}} when the action on tuples without repeated entries in <math>X^n</math> is sharply transitive.

==== Examples ==== The action of the symmetric group of {{math|''X''}} is transitive, in fact {{math|''n''}}-transitive for any {{math|''n''}} up to the cardinality of {{math|''X''}}. If {{math|''X''}} has cardinality {{math|''n''}}, the action of the [[alternating group]] is {{math|(''n'' − 2)}}-transitive but not {{math|(''n'' − 1)}}-transitive.

The action of the [[general linear group]] of a vector space {{math|''V''}} on the set {{math|''V'' &setminus; {{mset|0}}}} of non-zero vectors is transitive, but not 2-transitive (similarly for the action of the [[special linear group]] if the dimension of {{math|''v''}} is at least 2). The action of the [[orthogonal group]] of a Euclidean space is not transitive on nonzero vectors but it is on the [[unit sphere]].

=== Primitive actions === {{Main|primitive permutation group}} The action of {{math|''G''}} on {{math|''X''}} is called ''primitive'' if there is no [[Partition of a set|partition]] of {{math|''X''}} preserved by all elements of {{math|''G''}} apart from the trivial partitions (the partition in a single piece and its [[Dual space|dual]], the partition into [[Singleton (mathematics)|singletons]]).

=== Topological properties === Assume that <math>X</math> is a [[topological space]] and the action of <math>G</math> is by [[homeomorphism]]s.

The action is ''wandering'' if every <math>x\in X</math> has a [[Neighbourhood (mathematics)|neighbourhood]] <math>U</math> such that there are only finitely many <math>g\in G</math> with <math>(g\cdot U)\cap U\neq\emptyset</math>.{{sfn|Thurston|1997|loc=Definition 3.5.1(iv)}}

More generally, a point <math>x\in X</math> is called a point of discontinuity for the action of <math>G</math> if there is an open subset <math>U\ni x</math> such that there are only finitely many <math>g\in G</math> with <math>(g\cdot U)\cap U\neq\emptyset</math>. The ''domain of discontinuity'' of the action is the set of all points of discontinuity. Equivalently it is the largest <math>G</math>-stable open subset <math>\Omega\subset X</math> such that the action of <math>G</math> on <math>\Omega</math> is wandering.{{sfn|Kapovich|2009|loc=p. 73}} In a dynamical context this is also called a ''[[wandering set]]''.

The action is ''properly discontinuous'' if for every [[Compact space|compact]] subset <math>K\subset X</math> there are only finitely many <math>g\in G</math> such that <math>(g\cdot K)\cap K\neq\emptyset</math>. This is strictly stronger than wandering; for instance the action of <math>\mathbb{Z}</math> on <math>\mathbb{R}^2\backslash\{(0,0)\}</math> given by <math>n\cdot(x,y)=(2^nx,2^{-n}y)</math> is wandering and free but not properly discontinuous.{{sfn|Thurston|1980|p=176}}

The action by [[deck transformation]]s of the [[fundamental group]] of a locally [[simply connected space]] on a [[Covering space#Universal covering|universal cover]] is wandering and free. Such actions can be characterized by the following property: every <math>x\in X</math> has a neighbourhood <math>U</math> such that <math>(g\cdot U)\cap U=\emptyset</math> for every <math>g\in G\backslash\{e_G\}</math>.{{sfn|Hatcher|2002|loc=p. 72}} Actions with this property are sometimes called ''freely discontinuous'', and the largest subset on which the action is freely discontinuous is then called the ''free regular set''.{{sfn|Maskit|1988|loc=II.A.1, II.A.2}}

An action of a group <math>G</math> on a [[locally compact space]] <math>X</math> is called ''[[Cocompact group action|cocompact]]'' if there exists a compact subset <math>A\subset X</math> such that <math>X=G\cdot A</math>. For a properly discontinuous action, cocompactness is equivalent to compactness of the [[Quotient space (topology)|quotient space]] <math>X/G</math>.

=== Actions of topological groups === {{Main|Continuous group action}} Now assume <math>G</math> is a [[topological group]] and <math>X</math> a topological space on which it acts by homeomorphisms. The action is said to be ''continuous'' if the map <math>G\times X\rightarrow X</math> is continuous for the [[product topology]].

The action is said to be ''{{visible anchor|proper}}'' if the map <math>G\times X\rightarrow X\times X</math> defined by <math>(g,x)\mapsto(x,g\cdot x)</math> is [[proper map|proper]].{{sfn|tom Dieck|1987|loc=}} This means that given compact sets <math>K,K'</math> the set of <math>g\in G</math> such that <math>(g\cdot K)\cap K'\neq\emptyset</math> is compact. In particular, this is equivalent to proper discontinuity if <math>G</math> is a [[discrete group]].

It is said to be ''locally free'' if there exists a neighbourhood <math>U</math> of <math>e_G</math> such that <math>g\cdot x\neq x</math> for all <math>x\in X</math> and <math>g\in U\backslash\{e_G\}</math>.

The action is said to be ''strongly continuous'' if the orbital map <math>g\mapsto g\cdot x</math> is continuous for every <math>x\in X</math>. Contrary to what the name suggests, this is a weaker property than continuity of the action.{{citation needed|date=May 2023}}

If <math>G</math> is a [[Lie group]] and <math>X</math> a [[differentiable manifold]], then the subspace of ''smooth points'' for the action is the set of points <math>x\in X</math> such that the map <math>g\mapsto g\cdot x</math> is [[smooth map|smooth]]. There is a well-developed theory of [[Lie group action]]s, i.e. action which are smooth on the whole space.

=== Linear actions === {{Main|Group representation}} If {{math|''g''}} acts by [[Linear map|linear transformations]] on a [[Module (mathematics)|module]] over a [[commutative ring]], the action is said to be [[Irreducible representation|irreducible]] if there are no proper nonzero {{math|''g''}}-invariant submodules. It is said to be ''[[Semi-simplicity|semisimple]]'' if it decomposes as a [[direct sum]] of irreducible actions.

== Orbits and stabilizers ==  [[File:Compound of five tetrahedra.png|thumb|In the [[compound of five tetrahedra]], the symmetry group is the (rotational) icosahedral group {{math|''I''}} of order 60, while the stabilizer of a single chosen tetrahedron is the (rotational) [[tetrahedral group]] {{math|''T''}} of order 12, and the orbit space {{math|''I'' / ''T''}} (of order 60/12 = 5) is naturally identified with the 5 tetrahedra – the coset {{math|''gT''}} corresponds to the tetrahedron to which {{math|''g''}} sends the chosen tetrahedron.]]

Consider a group {{math|''G''}} acting on a set {{math|''X''}}. The ''{{visible anchor|orbit}}'' of an element {{math|''x''}} in {{math|''X''}} is the set of elements in {{math|''X''}} to which {{math|''x''}} can be moved by the elements of {{math|''G''}}. The orbit of {{math|''x''}} is denoted by {{math|''G''⋅''x''}}: <math display=block>G{\cdot}x = \{ g{\cdot}x : g \in G \}.</math>

The defining properties of a group guarantee that the set of orbits of (points {{math|''x''}} in) {{math|''X''}} under the action of {{math|''G''}} form a [[Partition of a set|partition]] of {{math|''X''}}. The associated [[equivalence relation]] is defined by saying {{math|''x'' ~ ''y''}} [[if and only if]] there exists a {{math|''g''}} in {{math|''G''}} with {{math|1=''g''⋅''x'' = ''y''}}. The orbits are then the [[equivalence class]]es under this relation; two elements {{math|''x''}} and {{math|''y''}} are equivalent if and only if their orbits are the same, that is, {{math|1=''G''⋅''x'' = ''G''⋅''y''}}.

The group action is [[#Notable properties of actions|transitive]] if and only if it has exactly one orbit, that is, if there exists {{math|''x''}} in {{math|''X''}} with {{math|1=''G''⋅''x'' = ''X''}}. This is the case if and only if {{math|1=''G''⋅''x'' = ''X''}} for {{em|all}} {{math|''x''}} in {{math|''X''}} (given that {{math|''X''}} is non-empty).

The set of all orbits of {{math|''X''}} under the action of {{math|''G''}} is written as {{math|''X'' / ''G''}} (or, less frequently, as {{math|''G'' \ ''X''}}), and is called the ''{{visible anchor|quotient}}'' of the action. In geometric situations it may be called the ''{{visible anchor|orbit space}}'', while in algebraic situations it may be called the space of ''{{visible anchor|coinvariants}}'', and written {{math|''X''''G''}}, by contrast with the invariants (fixed points), denoted {{math|''X''''G''}}: the coinvariants are a {{em|quotient}} while the invariants are a {{em|subset}}. The coinvariant terminology and notation are used particularly in [[group cohomology]] and [[group homology]], which use the same superscript/subscript convention.

=== Invariant subsets === If {{math|''Y''}} is a [[subset]] of {{math|''X''}}, then {{math|''G''⋅''Y''}} denotes the set {{math|{{mset|''g''⋅''y'' : ''g'' ∈ ''G'' and ''y'' ∈ ''Y''}}}}. The subset {{math|''Y''}} is said to be ''invariant under ''{{math|''G''}} if {{math|1=''G''⋅''Y'' = ''Y''}} (which is equivalent {{math|''G''⋅''Y'' ⊆ ''Y''}}). In that case, {{math|''G''}} also operates on {{math|''Y''}} by [[Restriction (mathematics)|restricting]] the action to {{math|''Y''}}. The subset {{math|''Y''}} is called ''fixed under ''{{math|''G''}} if {{math|1=''g''⋅''y'' = ''y''}} for all {{math|''g''}} in {{math|''G''}} and all {{math|''y''}} in {{math|''Y''}}. Every subset that is fixed under {{math|''G''}} is also invariant under {{math|''G''}}, but not conversely.

Every orbit is an invariant subset of {{math|''X''}} on which {{math|''G''}} acts [[#Notable properties of actions|transitively]]. Conversely, any invariant subset of {{math|''X''}} is a union of orbits. The action of {{math|''G''}} on {{math|''X''}} is ''transitive'' if and only if all elements are equivalent, meaning that there is only one orbit.

A {{math|''G''}}''-invariant'' element of {{math|''X''}} is {{math|''x'' ∈ ''X''}} such that {{math|1=''g''⋅''x'' = ''x''}} for all {{math|''g'' ∈ ''G''}}. The set of all such {{math|''x''}} is denoted {{math|''X''''G''}} and called the {{math|''G''}}''-invariants'' of {{math|''X''}}. When {{math|''X''}} is a [[G-module|{{math|''G''}}-module]], {{math|''X''''G''}} is the zeroth [[Group cohomology|cohomology]] group of {{math|''G''}} with coefficients in {{math|''X''}}, and the higher cohomology groups are the [[derived functor]]s of the [[functor]] of {{math|''G''}}-invariants.

=== Fixed points and stabilizer subgroups === Given {{math|''g''}} in {{math|''G''}} and {{math|''x''}} in {{math|''X''}} with {{math|1=''g''⋅''x'' = ''x''}}, it is said that "{{math|''x''}} is a fixed point of {{math|''g''}}" or that "{{math|''g''}} fixes {{math|''x''}}". For every {{math|''x''}} in {{math|''X''}}, the '''{{visible anchor|stabilizer subgroup}}''' of {{math|''G''}} with respect to {{math|''x''}} (also called the '''isotropy group''' or '''little group'''<ref name="Procesi">{{cite book|last1=Procesi|first1=Claudio|title=Lie Groups: An Approach through Invariants and Representations|date=2007|publisher=Springer Science & Business Media|isbn=9780387289298|page=5|url=https://books.google.com/books?id=Sl8OAGYRz_AC&q=%22little+group%22+action&pg=PA5|access-date=23 February 2017|language=en}}</ref>) is the set of all elements in {{math|''G''}} that fix {{math|''x''}}: <math display=block>G_x = \{g \in G : g{\cdot}x = x\}.</math> This is a [[subgroup]] of {{math|''G''}}, though typically not a normal one. The action of {{math|''G''}} on {{math|''X''}} is [[#Notable properties of actions|free]] if and only if all stabilizers are trivial. The kernel {{math|''N''}} of the homomorphism with the symmetric group, {{math|''G'' → Sym(''X'')}}, is given by the [[Intersection (set theory)|intersection]] of the stabilizers {{math|''G''''x''}} for all {{math|''x''}} in {{math|''X''}}. If {{math|''N''}} is trivial, the action is said to be faithful (or effective).

Let {{math|''x''}} and {{math|''y''}} be two elements in {{math|''X''}}, and let {{math|''g''}} be a group element such that {{math|1=''y'' = ''g''⋅''x''}}. Then the two stabilizer groups {{math|''G''''x''}} and {{math|''G''''y''}} are related by {{math|1=''G''''y'' = ''gG''''x''''g''−1}}.

Proof: by definition, {{math|''h'' ∈ ''G''''y''}} if and only if {{math|1=''h''⋅(''g''⋅''x'') = ''g''⋅''x''}}. Applying {{math|''g''−1}} to both sides of this equality yields {{math|1=(''g''−1''hg'')⋅''x'' = ''x''}}; that is, {{math|''g''−1''hg'' ∈ ''G''''x''}}.

An opposite inclusion follows similarly by taking {{math|''h'' ∈ ''G''''x''}} and {{math|1=''x'' = ''g''−1⋅''y''}}.

The above says that the stabilizers of elements in the same orbit are [[Conjugacy class|conjugate]] to each other. Thus, to each orbit, we can associate a [[conjugacy class]] of a subgroup of {{math|''G''}} (that is, the set of all conjugates of the subgroup). Let {{math|(''H'')}} denote the conjugacy class of {{math|''H''}}. Then the orbit {{math|''O''}} has type {{math|(''H'')}} if the stabilizer {{math|''G''''x''}} of some/any {{math|''x''}} in {{math|''O''}} belongs to {{math|(''H'')}}. A maximal orbit type is often called a [[principal orbit type]].

=== {{visible anchor|Orbit–stabilizer theorem}} === Orbits and stabilizers are closely related. For a fixed {{math|''x''}} in {{math|''X''}}, consider the map {{math|''f'' : ''G'' → ''X''}} given by {{math|''g'' ↦ ''g''⋅''x''}}. By definition the image {{math|''f''(''G'')}} of this map is the orbit {{math|''G''⋅''x''}}. The condition for two elements to have the same image is <math display=block>f(g)=f(h) \iff g{\cdot}x = h{\cdot}x \iff g^{-1}h{\cdot}x = x \iff g^{-1}h \in G_x \iff h \in gG_x.</math> In other words, {{math|1=''f''(''g'') = ''f''(''h'')}} ''if and only if'' {{math|''g''}} and {{math|''h''}} lie in the same [[coset]] for the stabilizer subgroup {{math|''G''''x''}}. Thus, the [[Fiber (mathematics)|fiber]] {{math|''f''{{i sup|−1}}({{mset|''y''}})}} of {{math|''f''}} over any {{math|''y''}} in {{math|''G''⋅''x''}} is contained in such a coset, and every such coset also occurs as a fiber. Therefore {{math|''f''}} induces a {{em|bijection}} between the set {{math|''G'' / ''G''''x''}} of cosets for the stabilizer subgroup and the orbit {{math|''G''⋅''x''}}, which sends {{math|''gG''''x'' ↦ ''g''⋅''x''}}.<ref>M. Artin, ''Algebra'', Proposition 6.8.4 on p. 179</ref> This result is known as the ''orbit–stabilizer theorem''.

If {{math|''G''}} is finite then the orbit–stabilizer theorem, together with [[Lagrange's theorem (group theory)|Lagrange's theorem]], gives <math display=block>|G \cdot x| = [G\,:\,G_x] = |G| / |G_x|.</math> In other words, the length of the orbit of {{math|''x''}} times the order of its stabilizer is the [[Order (group theory)|order of the group]]. In particular that implies that the orbit length is a divisor of the group order.

; Example: Let {{math|''G''}} be a group of prime order {{math|''p''}} acting on a set {{math|''X''}} with {{math|''k''}} elements. Since each orbit has either {{math|1}} or {{math|''p''}} elements, there are at least {{math|''k'' mod ''p''}} orbits of length {{math|1}} which are {{math|''G''}}-invariant elements. More specifically, {{math|''k''}} and the number of {{math|''G''}}-invariant elements are congruent modulo {{math|''p''}}.<ref>{{Cite book |last=Carter |first=Nathan |title=Visual Group Theory |publisher=The Mathematical Association of America |year=2009 |isbn=978-0883857571 |edition=1st |pages=200}}</ref>

This result is especially useful since it can be employed for counting arguments (typically in situations where {{math|''X''}} is finite as well).

[[File:Labeled cube graph.png|thumb|Cubical graph with vertices labeled]] ; Example: We can use the orbit–stabilizer theorem to count the automorphisms of a [[Graph (discrete mathematics)|graph]]. Consider the [[cubical graph]] as pictured, and let {{math|''G''}} denote its [[Graph automorphism|automorphism]] group. Then {{math|''G''}} acts on the set of vertices {{math|{{mset|1, 2, ..., 8}}}}, and this action is transitive as can be seen by composing rotations about the center of the cube. Thus, by the orbit–stabilizer theorem, {{math|1={{abs|''G''}} = {{abs|''G'' ⋅ 1}} {{abs|''G''1}} = 8 {{abs|''G''1}}}}. Applying the theorem now to the stabilizer {{math|''G''1}}, we can obtain {{math|1={{abs|''G''1}} = {{abs|(''G''1) ⋅ 2}} {{abs|(''G''1)2}}}}. Any element of {{math|''G''}} that fixes 1 must send 2 to either 2, 4, or 5. As an example of such automorphisms consider the rotation around the diagonal axis through 1 and 7 by {{math|2''π''/3}}, which permutes 2, 4, 5 and 3, 6, 8, and fixes 1 and 7. Thus, {{math|1={{abs|(''G''1) ⋅ 2}} = 3}}. Applying the theorem a third time gives {{math|1={{abs|(''G''1)2}} = {{abs|((''G''1)2) ⋅ 3}} {{abs|((''G''1)2)3}}}}. Any element of {{math|''G''}} that fixes 1 and 2 must send 3 to either 3 or 6. Reflecting the cube at the plane through 1, 2, 7 and 8 is such an automorphism sending 3 to 6, thus {{math|1={{abs|((''G''1)2) ⋅ 3}} = 2}}. One also sees that {{math|((''G''1)2)3}} consists only of the identity automorphism, as any element of {{math|''G''}} fixing 1, 2 and 3 must also fix all other vertices, since they are determined by their adjacency to 1, 2 and 3. Combining the preceding calculations, we can now obtain {{math|1={{abs|G}} = 8 ⋅ 3 ⋅ 2 ⋅ 1 = 48}}.

=== Burnside's lemma === A result closely related to the orbit–stabilizer theorem is [[Burnside's lemma]]: <math display=block>|X/G|=\frac{1}{|G|}\sum_{g\in G} |X^g|,</math> where {{math|''X''''g''}} is the set of points fixed by {{math|''g''}}. This result is mainly of use when {{math|''G''}} and {{math|''X''}} are finite, when it can be interpreted as follows: the number of orbits is equal to the average number of points fixed per group element.

Fixing a group {{math|''G''}}, the set of formal differences of finite {{math|''G''}}-sets forms a ring called the [[Burnside ring]] of {{math|''G''}}, where addition corresponds to [[disjoint union]], and multiplication to [[Cartesian product]].

== 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''−1}}. An exponential notation is commonly used for the right-action variant: {{math|1=''xg'' = ''g''−1''xg''}}; it satisfies ({{math|1=''x''''g'')''h'' = ''x''''gh''}}. * 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''−1}} 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''n''}} 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''1, ''x''2, ..., ''x''''n'') ↦ (''ax''1, ''ax''2, ..., ''ax''''n'')}} is a group action called [[scalar multiplication]]. * The automorphism group of a vector space (or [[graph theory|graph]], or group, or ring ...) 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''''n''}}. The group operations are given by multiplying the matrices from the groups with the vectors from {{math|''K''''n''}}. * The general linear group {{math|GL(''n'', '''Z''')}} acts on {{math|'''Z'''''n''}} by natural matrix action. The orbits of its action are classified by the [[greatest common divisor]] of coordinates of the vector in {{math|'''Z'''''n''}}. * 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'''n(''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|−''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''(''xet'')}}, {{math|''f''(''x'')''e''''t''}}, {{math|''f''(''x'' + ''t'')''et''}}, or {{math|''f''(''xe''''t'') + ''t''}}, but not {{math|''f''(''xet'' + ''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'''3}}: for any such quaternion {{math|1=''z'' = cos ''α''/2 + '''v''' sin ''α''/2}}, the mapping {{math|1=''f''('''x''') = ''z'''''x'''''z''*}} 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|''α'' : ''X'' × ''G'' → ''Y''}}, and with left {{math|''G''}}-action given by {{math|1=''g''⋅''α'' = ''α'' ∘ (id''X'' × –''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.

== Group actions and groupoids == {{Main|Groupoid#Group action}} The notion of group action can be encoded by the ''action [[groupoid]]'' {{math|1=''G''′ = ''G'' ⋉ ''X''}} associated to the group action. The stabilizers of the action are the vertex groups of the groupoid and the orbits of the action are its components.

== Morphisms and isomorphisms between ''G''-sets == If {{math|''X''}} and {{math|''Y''}} are two {{math|''G''}}-sets, a ''morphism'' from {{math|''X''}} to {{math|''Y''}} is a function {{math|''f'' : ''X'' → ''Y''}} such that {{math|1=''f''(''g''⋅''x'') = ''g''⋅''f''(''x'')}} for all {{math|''g''}} in {{math|''G''}} and all {{math|''x''}} in {{math|''X''}}. Morphisms of {{math|''G''}}-sets are also called ''[[equivariant map]]s'' or {{math|''G''}}-''maps''.

The composition of two morphisms is again a morphism. If a morphism {{math|''f''}} is bijective, then its inverse is also a morphism. In this case {{math|''f''}} is called an ''[[isomorphism]]'', and the two {{math|''G''}}-sets {{math|''X''}} and {{math|''Y''}} are called ''isomorphic''; for all practical purposes, isomorphic {{math|''G''}}-sets are indistinguishable.

Some example isomorphisms: * Every regular {{math|''G''}} action is isomorphic to the action of {{math|''G''}} on {{math|''G''}} given by left multiplication. * Every free {{math|''G''}} action is isomorphic to {{math|''G'' × ''S''}}, where {{math|''S''}} is some set and {{math|''G''}} acts on {{math|''G'' × ''S''}} by left multiplication on the first coordinate. ({{math|''S''}} can be taken to be the set of orbits {{math|''X'' / ''G''}}.) * Every transitive {{math|''G''}} action is isomorphic to left multiplication by {{math|''G''}} on the set of left cosets of some subgroup {{math|''H''}} of {{math|''G''}}. ({{math|''H''}} can be taken to be the stabilizer group of any element of the original {{math|''G''}}-set.)

With this notion of morphism, the collection of all {{math|''G''}}-sets forms a [[category theory|category]]; this category is a [[Grothendieck topos]] (in fact, assuming a classical [[metalogic]], this [[topos]] will even be Boolean).

== Variants and generalizations == We can also consider actions of [[monoid]]s on sets, by using the same two axioms as above. This does not define bijective maps and equivalence relations however. See [[semigroup action]].

Instead of actions on sets, we can define actions of groups and monoids on objects of an arbitrary category: start with an object {{math|''X''}} of some category, and then define an action on {{math|''X''}} as a monoid homomorphism into the monoid of [[endomorphisms]] of {{math|''X''}}. If {{math|''X''}} has an underlying set, then all definitions and facts stated above can be carried over. For example, if we take the category of vector spaces, we obtain [[group representation]]s in this fashion.

We can view a group {{math|''G''}} as a category with a single object in which every morphism is [[Inverse element|invertible]].<ref>{{harvp|Perrone|2024|pages=7-9}}</ref> A (left) group action is then nothing but a (covariant) [[functor]] from {{math|''G''}} to the [[category of sets]], and a group representation is a functor from {{math|''G''}} to the [[category of vector spaces]].<ref>{{harvp|Perrone|2024|pages=36-39}}</ref> A morphism between {{math|''G''}}-sets is then a [[natural transformation]] between the group action functors.<ref>{{harvp|Perrone|2024|pages=69-71}}</ref> In analogy, an action of a [[groupoid]] is a functor from the groupoid to the category of sets or to some other category.

In addition to [[continuous group action|continuous actions]] of topological groups on topological spaces, one also often considers [[Lie group action|smooth actions]] of Lie groups on [[manifold|smooth manifold]]s, regular actions of [[algebraic group]]s on [[algebraic variety|algebraic varieties]], and [[group-scheme action|actions]] of [[group scheme]]s on [[scheme (mathematics)|schemes]]. All of these are examples of [[group object]]s acting on objects of their respective category.

== Gallery == <gallery widths="200px" heights="180"> File:Octahedral-group-action.png|Orbit of a fundamental spherical triangle (marked in red) under action of the full octahedral group File:Icosahedral-group-action.png|Orbit of a fundamental spherical triangle (marked in red) under action of the full icosahedral group </gallery>

== See also == * [[Gain graph]] * [[Group with operators]] * [[Measurable group action]] * [[Monoid action]] * [[Young–Deruyts development]]

== Notes == {{notelist}}

== Citations == {{reflist}}

== References == * {{cite book|last1=Aschbacher|first1=Michael|author1-link=Michael Aschbacher|title=Finite Group Theory|publisher=Cambridge University Press|year=2000|mr=1777008 |isbn=978-0-521-78675-1 }} * {{cite book |first=David |last=Dummit |author2=Richard Foote |year=2003 |title=Abstract Algebra |edition=3rd |publisher=Wiley |isbn=0-471-43334-9}} * {{cite book |last1=Eie |first1=Minking |last2=Chang |first2=Shou-Te |title=A Course on Abstract Algebra |year=2010 |publisher=World Scientific |isbn=978-981-4271-88-2 }} * {{citation |first=Allen |last=Hatcher |author-link=Allen Hatcher |title=Algebraic Topology |url=https://pi.math.cornell.edu/~hatcher/AT/ATpage.html |year=2002 |publisher=Cambridge University Press |isbn=978-0-521-79540-1 |mr=1867354 }}. * {{cite book | first = Joseph | last = Rotman | year = 1995 | title = An Introduction to the Theory of Groups | others = Graduate Texts in Mathematics '''148''' | edition = 4th | publisher = Springer-Verlag | isbn = 0-387-94285-8 }} * {{cite book |last1=Smith |first1=Jonathan D.H. |title=Introduction to abstract algebra |series=Textbooks in mathematics |year=2008 |publisher=CRC Press |isbn=978-1-4200-6371-4 }} * {{citation | last=Kapovich | first=Michael | title=Hyperbolic manifolds and discrete groups | zbl=1180.57001 | series=Modern Birkhäuser Classics | publisher=Birkhäuser | isbn=978-0-8176-4912-8 | pages=xxvii+467 | year=2009 }} * {{citation | last=Maskit | first=Bernard | title=Kleinian groups | zbl=0627.30039 | series=Grundlehren der Mathematischen Wissenschaften | volume=287 | publisher=Springer-Verlag |pages= XIII+326 | year=1988 }} * {{citation|last = Perrone |first = Paolo |title = Starting Category Theory |date = 2024 |publisher = World Scientific|doi = 10.1142/9789811286018_0005 |isbn = 978-981-12-8600-1 }} * {{citation |last1=Thurston |first1=William |title=The geometry and topology of three-manifolds |url=http://library.msri.org/books/gt3m/ |series=Princeton lecture notes |year=1980 |page=175 |access-date=2016-02-08 |archive-date=2020-07-27 |archive-url=https://web.archive.org/web/20200727020107/http://library.msri.org/books/gt3m/ |url-status=dead }} * {{citation | last=Thurston | first=William P. | title=Three-dimensional geometry and topology. Vol. 1. | zbl=0873.57001 | series=Princeton Mathematical Series | volume=35 | publisher=Princeton University Press | pages=x+311 | year=1997 }} * {{citation | last1=tom Dieck | first1=Tammo | title=Transformation groups | url=https://books.google.com/books?id=azcQhi6XeioC | publisher=Walter de Gruyter & Co. | location=Berlin | series=de Gruyter Studies in Mathematics | isbn=978-3-11-009745-0 | mr=889050 | year=1987 | volume=8 | page=29 | doi=10.1515/9783110858372.312 | url-access=subscription }}

== External links == * {{springer|title=Action of a group on a manifold|id=p/a010550}} * {{MathWorld|urlname=GroupAction|title=Group Action}}

[[Category:Group theory]] [[Category:Group actions| ]] [[Category:Representation theory of groups]] [[Category:Symmetry]]