Group action: Difference between revisions
Just made the proof part in the "Fixed points and stabilizer subgroups" section a little bit easier to read. |
imported>Georg-Johann m →{{visible anchor|Orbit–stabilizer theorem}}: Typo: Examples |
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[[File:Group action on equilateral triangle.svg|right|thumb|The [[cyclic group]] {{math|C<sub>3</sub>}} consisting of the [[Rotation (mathematics)|rotations]] by 0°, 120° and 240° acts on the set of the three vertices.]] | [[File:Group action on equilateral triangle.svg|right|thumb|The [[cyclic group]] {{math|C<sub>3</sub>}} consisting of the [[Rotation (mathematics)|rotations]] by 0°, 120° and 240° acts on the set of the three vertices.]] | ||
In | 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 | 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. | 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. | ||
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=== Left group action === | === 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 | 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> | : <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> | 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> | ||
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|<math>\alpha(g,\alpha(h,x))=\alpha(gh,x)</math> | |<math>\alpha(g,\alpha(h,x))=\alpha(gh,x)</math> | ||
|} | |} | ||
for all | 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''. | 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]] | 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 | 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>. | 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, | 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 | 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 === | === Right group action === | ||
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for all {{mvar|g}} and {{mvar|h}} in {{mvar|G}} and all {{mvar|x}} in {{mvar|X}}. | 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}}. | 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 a single group action, it suffices to consider only left actions. | ||
Thus, for establishing general properties of group | |||
== Notable properties of actions == | == Notable properties of actions == | ||
Let | 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 | 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 | 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 | 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 === | === Transitivity properties === | ||
The action of | 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 | 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 | 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 | 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 ==== | ==== Examples ==== | ||
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=== Topological properties === | === Topological properties === | ||
Assume that | 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 | 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 | 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 | 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 | 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 | 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 === | === Actions of topological groups === | ||
{{Main|Continuous group action}} | {{Main|Continuous group action}} | ||
Now assume | 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 | 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 | 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 | 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 | 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 === | === Linear actions === | ||
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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 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 [[ | 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''<sub>''G''</sub>}}, by contrast with the invariants (fixed points), denoted {{math|''X''<sup>''G''</sup>}}: 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. | 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''<sub>''G''</sub>}}, by contrast with the invariants (fixed points), denoted {{math|''X''<sup>''G''</sup>}}: 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. | ||
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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. | 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 [[ | 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''<sup>''G''</sup>}} and called the {{math|''G''}}''-invariants'' of {{math|''X''}}. When {{math|''X''}} is a [[G-module|{{math|''G''}}-module]], {{math|''X''<sup>''G''</sup>}} 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. | 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''<sup>''G''</sup>}} and called the {{math|''G''}}''-invariants'' of {{math|''X''}}. When {{math|''X''}} is a [[G-module|{{math|''G''}}-module]], {{math|''X''<sup>''G''</sup>}} 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. | ||
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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''}}: | 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> | <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 [[ | 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''<sub>''x''</sub>}} 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''<sub>''x''</sub>}} and {{math|''G''<sub>''y''</sub>}} are related by {{math|1=''G''<sub>''y''</sub> = ''gG''<sub>''x''</sub>''g''<sup>−1</sup>}}. | 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''<sub>''x''</sub>}} and {{math|''G''<sub>''y''</sub>}} are related by {{math|1=''G''<sub>''y''</sub> = ''gG''<sub>''x''</sub>''g''<sup>−1</sup>}}. | ||
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If {{math|''G''}} is finite then the orbit–stabilizer theorem, together with [[Lagrange's theorem (group theory)|Lagrange's theorem]], gives | 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 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). | 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]] | [[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''<sub>1</sub>}} = 8 {{abs|''G''<sub>1</sub>}}}}. Applying the theorem now to the stabilizer {{math|''G''<sub>1</sub>}}, we can obtain {{math|1={{abs|''G''<sub>1</sub>}} = {{abs|(''G''<sub>1</sub>) ⋅ 2}} {{abs|(''G''<sub>1</sub>)<sub>2</sub>}}}}. 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''<sub>1</sub>) ⋅ 2}} = 3}}. Applying the theorem a third time gives {{math|1={{abs|(''G''<sub>1</sub>)<sub>2</sub>}} = {{abs|((''G''<sub>1</sub>)<sub>2</sub>) ⋅ 3}} {{abs|((''G''<sub>1</sub>)<sub>2</sub>)<sub>3</sub>}}}}. 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''<sub>1</sub>)<sub>2</sub>) ⋅ 3}} = 2}}. One also sees that {{math|((''G''<sub>1</sub>)<sub>2</sub>)<sub>3</sub>}} 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 === | === Burnside's lemma === | ||
A result closely related to the | 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> | <math display=block>|X/G|=\frac{1}{|G|}\sum_{g\in G} |X^g|,</math> | ||
where {{math|''X''<sup>''g''</sup>}} 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. | where {{math|''X''<sup>''g''</sup>}} 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. | ||
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== Gallery == | == Gallery == | ||
<gallery widths="200px" heights="180"> | <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: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 | File:Icosahedral-group-action.png|Orbit of a fundamental spherical triangle (marked in red) under action of the full icosahedral group | ||
</gallery> | </gallery> | ||
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* {{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 |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 }} | * {{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= | * {{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 | * {{cite book | ||
| first = Joseph | | first = Joseph | ||
Latest revision as of 14:25, 14 May 2026
In mathematics, an action of a group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} on a set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is, loosely speaking, an operation that takes an element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} and an element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} and produces another element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S.} More formally, it is a group homomorphism from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} to the automorphism group of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} (the set of all bijections on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} along with group operation being function composition). One says that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} acts on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S.}
Many sets of transformations form a group under function composition; for example, the rotations 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 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 isometries acts on Euclidean space and also on the figures drawn in it; in particular, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron.
A group action on a vector space is called a 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{GL}(n,K)} , the group of the invertible matrices of dimension Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} over a field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} .
The symmetric group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_n} acts on any set with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} elements by permuting the elements of the set. Although the group of all permutations 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is a group with identity element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is a set, then a (left) group action Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is a function
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha : G \times X \to X}
that satisfies the following two axioms:[1]
Identity: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha(e,x)=x} Compatibility: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha(g,\alpha(h,x))=\alpha(gh,x)}
for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} and all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} .
The group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is then said to act on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} (from the left). A set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} together with an action of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is called a (left) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} -set.
It can be notationally convenient to curry the action Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} , so that, instead, one has a collection of transformations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha_g:X\rightarrow X} , with one transformation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha_g} for each group element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\in G} . The identity and compatibility relations then read Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha_e(x) = x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha_g(\alpha_h(x)) = (\alpha_g \circ \alpha_h)(x) = \alpha_{gh}(x)} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha_g\circ\alpha_h=\alpha_{gh}} .
With the above understanding, it is very common to avoid writing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} entirely, and to replace it with either a dot, or with nothing at all. Thus, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha(g,x)} can be shortened to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\cdot x} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle gx} , especially when the action is clear from context. The axioms are then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\{\begin{align}& e\cdot x=x \\ & g\cdot(h\cdot x)=(gh)\cdot x\end{align}\right.}
From these two axioms, it follows that for any fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} , the function from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} to itself which maps Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\cdot x} is a bijection, with inverse bijection the corresponding map for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g^{-1}} . Therefore, one may equivalently define a group action of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} as a group homomorphism from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} into the symmetric group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Sym}(X)} of all bijections from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} to itself.[2]
Right group action
Likewise, a right group action of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is a function
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha : X \times G \to X,}
that satisfies the analogous axioms:[3]
Identity: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha(x,e)=x} Compatibility: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha(\alpha(x,g),h)=\alpha(x,gh)}
(with α(x, g) often shortened to xg or x⋅g when the action being considered is clear from context)
Identity: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x{\cdot}e = x} Compatibility: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x{\cdot}g){\cdot}h = x{\cdot}(gh)}
for all g and h in G and all x in X.
The difference between left and right actions is in the order in which a product gh acts on x. For a left action, h acts first, followed by g second. For a right action, g acts first, followed by h second. Because of the formula (gh)−1 = h−1g−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 G on X can be considered as a left action of its opposite group Gop on X. Thus, for establishing general properties of a single group action, it suffices to consider only left actions.
Notable properties of actions
Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} be a group acting on a set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} . The action is called faithful or effective if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\cdot x=x} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in X} implies that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g=e_G} . Equivalently, the homomorphism from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} to the group of bijections of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} corresponding to the action is injective.
The action is called free (or semiregular or fixed-point free) if the statement that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\cdot x=x} for some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in X} already implies that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g=e_G} . In other words, no non-trivial element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} fixes a point of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} . 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 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathbb{Z}/2\mathbb{Z})^n} (of cardinality Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^n} ) acts faithfully on a set of size Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2n} . This is not always the case, for example the cyclic group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}/2^n\mathbb{Z}} cannot act faithfully on a set of size less than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^n} .
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_5} , the icosahedral group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_5\times\mathbb{Z}/2\mathbb{Z}} and the cyclic group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}/120\mathbb{Z}} . 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is called transitive if for any two points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,y\in X} there exists a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\in G} so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\cdot x=y} .
The action is simply transitive (or sharply transitive, or regular) if it is both transitive and free. This means that given Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,y\in X} there is exactly one Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\in G} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\cdot x=y} . If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is acted upon simply transitively by a group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} then it is called a principal homogeneous space for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} or a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} -torsor.
For an integer Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\geq 1} , the action is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -transitive if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} has at least Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} elements, and for any pair of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -tuples Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_1,\ldots,x_n),(y_1,\ldots,y_n)\in X^n} with pairwise distinct entries (that is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i\neq x_j} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_i\neq y_j} when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i\neq j} ) there exists a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\in G} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\cdot x_i=y_i} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=1,\ldots,n} . In other words, the action on the subset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X^n} of tuples without repeated entries is transitive. For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=2,3} this is often called double, respectively triple, transitivity. The class of 2-transitive groups (that is, subgroups of a finite symmetric group whose action is 2-transitive) and more generally multiply transitive groups is well-studied in finite group theory.
An action is sharply Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -transitive when the action on tuples without repeated entries in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X^n} is sharply transitive.
Examples
The action of the symmetric group of X is transitive, in fact n-transitive for any n up to the cardinality of X. If X has cardinality n, the action of the alternating group is (n − 2)-transitive but not (n − 1)-transitive.
The action of the general linear group of a vector space V on the set V ∖ {0} of non-zero vectors is transitive, but not 2-transitive (similarly for the action of the special linear group if the dimension of 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
The action of G on X is called primitive if there is no partition of X preserved by all elements of G apart from the trivial partitions (the partition in a single piece and its dual, the partition into singletons).
Topological properties
Assume that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is a topological space and the action of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is by homeomorphisms.
The action is wandering if every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in X} has a neighbourhood Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} such that there are only finitely many Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\in G} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (g\cdot U)\cap U\neq\emptyset} .[4]
More generally, a point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in X} is called a point of discontinuity for the action of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} if there is an open subset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U\ni x} such that there are only finitely many Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\in G} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (g\cdot U)\cap U\neq\emptyset} . The domain of discontinuity of the action is the set of all points of discontinuity. Equivalently it is the largest Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} -stable open subset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega\subset X} such that the action of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} is wandering.[5] In a dynamical context this is also called a wandering set.
The action is properly discontinuous if for every compact subset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K\subset X} there are only finitely many Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\in G} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (g\cdot K)\cap K\neq\emptyset} . This is strictly stronger than wandering; for instance the action of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^2\backslash\{(0,0)\}} given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\cdot(x,y)=(2^nx,2^{-n}y)} is wandering and free but not properly discontinuous.[6]
The action by deck transformations of the fundamental group of a locally simply connected space on a universal cover is wandering and free. Such actions can be characterized by the following property: every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in X} has a neighbourhood Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (g\cdot U)\cap U=\emptyset} for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\in G\backslash\{e_G\}} .[7] 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.[8]
An action of a group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} on a locally compact space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is called cocompact if there exists a compact subset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\subset X} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X=G\cdot A} . For a properly discontinuous action, cocompactness is equivalent to compactness of the quotient space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X/G} .
Actions of topological groups
Now assume Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is a topological group and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} a topological space on which it acts by homeomorphisms. The action is said to be continuous if the map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G\times X\rightarrow X} is continuous for the product topology.
The action is said to be proper if the map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G\times X\rightarrow X\times X} defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (g,x)\mapsto(x,g\cdot x)} is proper.[9] This means that given compact sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K,K'} the set of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\in G} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (g\cdot K)\cap K'\neq\emptyset} is compact. In particular, this is equivalent to proper discontinuity if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is a discrete group.
It is said to be locally free if there exists a neighbourhood Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_G} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\cdot x\neq x} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\in U\backslash\{e_G\}} .
The action is said to be strongly continuous if the orbital map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\mapsto g\cdot x} is continuous for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in X} . Contrary to what the name suggests, this is a weaker property than continuity of the action.[citation needed]
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is a Lie group and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} a differentiable manifold, then the subspace of smooth points for the action is the set of points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in X} such that the map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\mapsto g\cdot x} is smooth. There is a well-developed theory of Lie group actions, i.e. action which are smooth on the whole space.
Linear actions
If g acts by linear transformations on a module over a commutative ring, the action is said to be irreducible if there are no proper nonzero g-invariant submodules. It is said to be semisimple if it decomposes as a direct sum of irreducible actions.
Orbits and stabilizers
Consider a group G acting on a set X. The orbit of an element x in X is the set of elements in X to which x can be moved by the elements of G. The orbit of x is denoted by G⋅x: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G{\cdot}x = \{ g{\cdot}x : g \in G \}.}
The defining properties of a group guarantee that the set of orbits of (points x in) X under the action of G form a partition of X. The associated equivalence relation is defined by saying x ~ y if and only if there exists a g in G with g⋅x = y. The orbits are then the equivalence classes under this relation; two elements x and y are equivalent if and only if their orbits are the same, that is, G⋅x = G⋅y.
The group action is transitive if and only if it has exactly one orbit, that is, if there exists x in X with G⋅x = X. This is the case if and only if G⋅x = X for all x in X (given that X is non-empty).
The set of all orbits of X under the action of G is written as X / G (or, less frequently, as G \ X), and is called the quotient of the action. In geometric situations it may be called the orbit space, while in algebraic situations it may be called the space of coinvariants, and written XG, by contrast with the invariants (fixed points), denoted XG: the coinvariants are a quotient while the invariants are a 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 Y is a subset of X, then G⋅Y denotes the set {g⋅y : g ∈ G and y ∈ Y}. The subset Y is said to be invariant under G if G⋅Y = Y (which is equivalent G⋅Y ⊆ Y). In that case, G also operates on Y by restricting the action to Y. The subset Y is called fixed under G if g⋅y = y for all g in G and all y in Y. Every subset that is fixed under G is also invariant under G, but not conversely.
Every orbit is an invariant subset of X on which G acts transitively. Conversely, any invariant subset of X is a union of orbits. The action of G on X is transitive if and only if all elements are equivalent, meaning that there is only one orbit.
A G-invariant element of X is x ∈ X such that g⋅x = x for all g ∈ G. The set of all such x is denoted XG and called the G-invariants of X. When X is a G-module, XG is the zeroth cohomology group of G with coefficients in X, and the higher cohomology groups are the derived functors of the functor of G-invariants.
Fixed points and stabilizer subgroups
Given g in G and x in X with g⋅x = x, it is said that "x is a fixed point of g" or that "g fixes x". For every x in X, the stabilizer subgroup of G with respect to x (also called the isotropy group or little group[10]) is the set of all elements in G that fix x: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_x = \{g \in G : g{\cdot}x = x\}.} This is a subgroup of G, though typically not a normal one. The action of G on X is free if and only if all stabilizers are trivial. The kernel N of the homomorphism with the symmetric group, G → Sym(X), is given by the intersection of the stabilizers Gx for all x in X. If N is trivial, the action is said to be faithful (or effective).
Let x and y be two elements in X, and let g be a group element such that y = g⋅x. Then the two stabilizer groups Gx and Gy are related by Gy = gGxg−1.
Proof: by definition, h ∈ Gy if and only if h⋅(g⋅x) = g⋅x. Applying g−1 to both sides of this equality yields (g−1hg)⋅x = x; that is, g−1hg ∈ Gx.
An opposite inclusion follows similarly by taking h ∈ Gx and x = g−1⋅y.
The above says that the stabilizers of elements in the same orbit are conjugate to each other. Thus, to each orbit, we can associate a conjugacy class of a subgroup of G (that is, the set of all conjugates of the subgroup). Let (H) denote the conjugacy class of H. Then the orbit O has type (H) if the stabilizer Gx of some/any x in O belongs to (H). A maximal orbit type is often called a principal orbit type.
Orbit–stabilizer theorem
Orbits and stabilizers are closely related. For a fixed x in X, consider the map f : G → X given by g ↦ g⋅x. By definition the image f(G) of this map is the orbit G⋅x. The condition for two elements to have the same image is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.} In other words, f(g) = f(h) if and only if g and h lie in the same coset for the stabilizer subgroup Gx. Thus, the fiber f−1({y}) of f over any y in G⋅x is contained in such a coset, and every such coset also occurs as a fiber. Therefore f induces a bijection between the set G / Gx of cosets for the stabilizer subgroup and the orbit G⋅x, which sends gGx ↦ g⋅x.[11] This result is known as the orbit–stabilizer theorem.
If G is finite then the orbit–stabilizer theorem, together with Lagrange's theorem, gives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |G \cdot x| = [G\,:\,G_x] = |G| / |G_x|.} In other words, the length of the orbit of x times the order of its stabilizer is the order of the group. In particular that implies that the orbit length is a divisor of the group order.
- Example
- Let G be a group of prime order p acting on a set X with k elements. Since each orbit has either 1 or p elements, there are at least k mod p orbits of length 1 which are G-invariant elements. More specifically, k and the number of G-invariant elements are congruent modulo p.[12]
This result is especially useful since it can be employed for counting arguments (typically in situations where X is finite as well).
- Example
- We can use the orbit–stabilizer theorem to count the automorphisms of a graph. Consider the cubical graph as pictured, and let G denote its automorphism group. Then G acts on the set of vertices {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, |G| = |G ⋅ 1| |G1| = 8 |G1|. Applying the theorem now to the stabilizer G1, we can obtain |G1| = |(G1) ⋅ 2| |(G1)2|. Any element of 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 2π/3, which permutes 2, 4, 5 and 3, 6, 8, and fixes 1 and 7. Thus, |(G1) ⋅ 2| = 3. Applying the theorem a third time gives |(G1)2| = |((G1)2) ⋅ 3| |((G1)2)3|. Any element of 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 |((G1)2) ⋅ 3| = 2. One also sees that ((G1)2)3 consists only of the identity automorphism, as any element of 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 |G| = 8 ⋅ 3 ⋅ 2 ⋅ 1 = 48.
Burnside's lemma
A result closely related to the orbit–stabilizer theorem is Burnside's lemma: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |X/G|=\frac{1}{|G|}\sum_{g\in G} |X^g|,} where Xg is the set of points fixed by g. This result is mainly of use when G and 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 G, the set of formal differences of finite G-sets forms a ring called the Burnside ring of G, where addition corresponds to disjoint union, and multiplication to Cartesian product.
Examples
- The trivial action of any group G on any set X is defined by g⋅x = x for all g in G and all x in X; that is, every group element induces the identity permutation on X.[13]
- In every group G, left multiplication is an action of G on G: g⋅x = gx for all g, x in 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 G.
- In every group G with subgroup H, left multiplication is an action of G on the set of cosets G / H: g⋅aH = gaH for all g, a in G. In particular if H contains no nontrivial normal subgroups of G this induces an isomorphism from G to a subgroup of the permutation group of degree [G : H].
- In every group G, conjugation is an action of G on G: g⋅x = gxg−1. An exponential notation is commonly used for the right-action variant: xg = g−1xg; it satisfies (xg)h = xgh.
- In every group G with subgroup H, conjugation is an action of G on conjugates of H: g⋅K = gKg−1 for all g in G and K conjugates of H.
- An action of Z on a set X uniquely determines and is determined by an automorphism of X, given by the action of 1. Similarly, an action of Z / 2Z on X is equivalent to the data of an involution of X.
- The symmetric group Sn and its subgroups act on the set {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 V over a field F with group of units F*, the mapping F* × V → V given by a × (x1, x2, ..., xn) ↦ (ax1, ax2, ..., axn) is a group action called scalar multiplication.
- The automorphism group of a vector space (or 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 GL(n, K) and its subgroups, particularly its Lie subgroups (including the special linear group SL(n, K), orthogonal group O(n, K), special orthogonal group SO(n, K), and symplectic group Sp(n, K)) are Lie groups that act on the vector space Kn. The group operations are given by multiplying the matrices from the groups with the vectors from Kn.
- The general linear group GL(n, Z) acts on Zn by natural matrix action. The orbits of its action are classified by the greatest common divisor of coordinates of the vector in Zn.
- The affine group acts 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;[14] indeed this can be used to give a definition of an affine space.
- The projective linear group PGL(n + 1, K) and its subgroups, particularly its Lie subgroups, which are Lie groups that act on the projective space Pn(K). This is a quotient of the action of the general linear group on projective space. Particularly notable is PGL(2, K), the symmetries of the projective line, which is sharply 3-transitive, preserving the cross ratio; the Möbius group PGL(2, C) is of particular interest.
- The isometries of the plane act on the set of 2D images and patterns, such as wallpaper patterns. 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 ]
- The sets acted on by a group G comprise the category of G-sets in which the objects are G-sets and the morphisms are G-set homomorphisms: functions f : X → Y such that g⋅(f(x)) = f(g⋅x) for every g in G.
- The Galois group of a field extension L / K acts on the field L but has only a trivial action on elements of the subfield K. Subgroups of Gal(L / K) correspond to subfields of L that contain K, that is, intermediate field extensions between L and K.
- The additive group of the real numbers (R, +) acts on the phase space of "well-behaved" systems in classical mechanics (and in more general dynamical systems) by time translation: if t is in R and x is in the phase space, then x describes a state of the system, and t + x is defined to be the state of the system t seconds later if t is positive or −t seconds ago if t is negative.
- The additive group of the real numbers (R, +) acts on the set of real functions of a real variable in various ways, with (t⋅f)(x) equal to, for example, f(x + t), f(x) + t, f(xet), f(x)et, f(x + t)et, or f(xet) + t, but not f(xet + t).
- Given a group action of G on X, we can define an induced action of G on the power set of X, by setting g⋅U = {g⋅u : u ∈ U} for every subset U of X and every g in 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 geometries.
- The quaternions with norm 1 (the versors), as a multiplicative group, act on R3: for any such quaternion z = cos α/2 + v sin α/2, the mapping f(x) = zxz* is a counterclockwise rotation through an angle α about an axis given by a unit vector v; z is the same rotation; see quaternions and spatial rotation. This is not a faithful action because the quaternion −1 leaves all points where they were, as does the quaternion 1.
- Given left G-sets X, Y, there is a left G-set YX whose elements are G-equivariant maps α : X × G → Y, and with left G-action given by g⋅α = α ∘ (idX × –g) (where "–g" indicates right multiplication by g). This G-set has the property that its fixed points correspond to equivariant maps X → Y; more generally, it is an exponential object in the category of G-sets.
Group actions and groupoids
The notion of group action can be encoded by the action groupoid 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 X and Y are two G-sets, a morphism from X to Y is a function f : X → Y such that f(g⋅x) = g⋅f(x) for all g in G and all x in X. Morphisms of G-sets are also called equivariant maps or G-maps.
The composition of two morphisms is again a morphism. If a morphism f is bijective, then its inverse is also a morphism. In this case f is called an isomorphism, and the two G-sets X and Y are called isomorphic; for all practical purposes, isomorphic G-sets are indistinguishable.
Some example isomorphisms:
- Every regular G action is isomorphic to the action of G on G given by left multiplication.
- Every free G action is isomorphic to G × S, where S is some set and G acts on G × S by left multiplication on the first coordinate. (S can be taken to be the set of orbits X / G.)
- Every transitive G action is isomorphic to left multiplication by G on the set of left cosets of some subgroup H of G. (H can be taken to be the stabilizer group of any element of the original G-set.)
With this notion of morphism, the collection of all G-sets forms a 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 monoids 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 X of some category, and then define an action on X as a monoid homomorphism into the monoid of endomorphisms of X. If 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 representations in this fashion.
We can view a group G as a category with a single object in which every morphism is invertible.[15] A (left) group action is then nothing but a (covariant) functor from G to the category of sets, and a group representation is a functor from G to the category of vector spaces.[16] A morphism between G-sets is then a natural transformation between the group action functors.[17] 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 actions of topological groups on topological spaces, one also often considers smooth actions of Lie groups on smooth manifolds, regular actions of algebraic groups on algebraic varieties, and actions of group schemes on schemes. All of these are examples of group objects acting on objects of their respective category.
Gallery
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Orbit of a fundamental spherical triangle (marked in red) under action of the full octahedral group
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Orbit of a fundamental spherical triangle (marked in red) under action of the full icosahedral groupOrbit of a fundamental spherical triangle (marked in red) under action of the full icosahedral group
See also
Notes
Citations
- ↑ Eie & Chang (2010). [[[:Template:Google books]] A Course on Abstract Algebra] Check
|url=value (help). p. 144. - ↑ This is done, for example, by Smith (2008). [[[:Template:Google books]] Introduction to abstract algebra] Check
|url=value (help). p. 253. - ↑ "Definition:Right Group Action Axioms". Proof Wiki. Retrieved 19 December 2021.
- ↑ Thurston 1997, Definition 3.5.1(iv).
- ↑ Kapovich 2009, p. 73.
- ↑ Thurston 1980, p. 176.
- ↑ Hatcher 2002, p. 72.
- ↑ Maskit 1988, II.A.1, II.A.2.
- ↑ tom Dieck 1987.
- ↑ Procesi, Claudio (2007). Lie Groups: An Approach through Invariants and Representations. Springer Science & Business Media. p. 5. ISBN 9780387289298. Retrieved 23 February 2017.
- ↑ M. Artin, Algebra, Proposition 6.8.4 on p. 179
- ↑ Carter, Nathan (2009). Visual Group Theory (1st ed.). The Mathematical Association of America. p. 200. ISBN 978-0883857571.
- ↑ Eie & Chang (2010). [[[:Template:Google books]] A Course on Abstract Algebra] Check
|url=value (help). p. 145. - ↑ Reid, Miles (2005). Geometry and topology. Cambridge, UK New York: Cambridge University Press. p. 170. ISBN 9780521613255.
- ↑ Perrone (2024), pp. 7–9
- ↑ Perrone (2024), pp. 36–39
- ↑ Perrone (2024), pp. 69–71
References
- Aschbacher, Michael (2000). Finite Group Theory. Cambridge University Press. ISBN 978-0-521-78675-1. MR 1777008.
- Dummit, David; Richard Foote (2003). Abstract Algebra (3rd ed.). Wiley. ISBN 0-471-43334-9.
- Eie, Minking; Chang, Shou-Te (2010). A Course on Abstract Algebra. World Scientific. ISBN 978-981-4271-88-2.
- Hatcher, Allen (2002), Algebraic Topology, Cambridge University Press, ISBN 978-0-521-79540-1, MR 1867354.
- Rotman, Joseph (1995). An Introduction to the Theory of Groups. Graduate Texts in Mathematics 148 (4th ed.). Springer-Verlag. ISBN 0-387-94285-8.
- Smith, Jonathan D.H. (2008). Introduction to abstract algebra. Textbooks in mathematics. CRC Press. ISBN 978-1-4200-6371-4.
- Kapovich, Michael (2009), Hyperbolic manifolds and discrete groups, Modern Birkhäuser Classics, Birkhäuser, pp. xxvii+467, ISBN 978-0-8176-4912-8, Zbl 1180.57001
- Maskit, Bernard (1988), Kleinian groups, Grundlehren der Mathematischen Wissenschaften, 287, Springer-Verlag, pp. XIII+326, Zbl 0627.30039
- Perrone, Paolo (2024), Starting Category Theory, World Scientific, doi:10.1142/9789811286018_0005, ISBN 978-981-12-8600-1
- Thurston, William (1980), The geometry and topology of three-manifolds, Princeton lecture notes, p. 175, archived from the original on 2020-07-27, retrieved 2016-02-08
- Thurston, William P. (1997), Three-dimensional geometry and topology. Vol. 1., Princeton Mathematical Series, 35, Princeton University Press, pp. x+311, Zbl 0873.57001
- tom Dieck, Tammo (1987), Transformation groups, de Gruyter Studies in Mathematics, 8, Berlin: Walter de Gruyter & Co., p. 29, doi:10.1515/9783110858372.312, ISBN 978-3-11-009745-0, MR 0889050