Editing Group action (section)

== Notable properties of actions ==
Let {{math|''G''}} be a group acting on a set {{math|''X''}}. The action is called ''{{visible anchor|faithful}}'' or ''{{visible anchor|effective}}'' if {{math|1=''g''⋅''x'' = ''x''}} for all {{math|''x'' ∈ ''X''}} implies that {{math|1=''g'' = ''e''<sub>''G''</sub>}}. Equivalently, the [[homomorphism]] from {{math|''G''}} to the group of bijections of {{math|''X''}} corresponding to the action is [[injective]].

The action is called ''{{visible anchor|free}}'' (or ''semiregular'' or ''fixed-point free'') if the statement that {{math|1=''g''&sdot;''x'' = ''x''}} for some {{math|''x'' ∈ ''X''}} already implies that {{math|1=''g'' = ''e''<sub>''G''</sub>}}. In other words, no non-trivial element of {{math|''G''}} fixes a point of {{math|''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 [[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|('''Z''' / 2'''Z''')<sup>''n''</sup>}} (of cardinality {{math|2<sup>''n''</sup>}}) acts faithfully on a set of size {{math|2''n''}}. This is not always the case, for example the [[cyclic group]] {{math|'''Z''' / 2<sup>''n''</sup>'''Z'''}} cannot act faithfully on a set of size less than {{math|2<sup>''n''</sup>}}.

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<sub>5</sub>}}, the icosahedral group {{math|A<sub>5</sub> × '''Z''' / 2'''Z'''}} and the cyclic group {{math|'''Z''' / 120'''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 {{math|''G''}} on {{math|''X''}} is called ''{{visible anchor|transitive}}'' if for any two points {{math|''x'', ''y'' ∈ ''X''}} there exists a {{math|''g'' ∈ ''G''}} so that {{math|1=''g'' &sdot; ''x'' = ''y''}}.

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'' ∈ ''X''}} there is exactly one {{math|''g'' ∈ ''G''}} such that {{math|1=''g'' &sdot; ''x'' = ''y''}}. If {{math|''X''}} is acted upon simply transitively by a group {{math|''G''}} then it is called a [[principal homogeneous space]] for {{math|''G''}} or a {{math|''G''}}-torsor.

For an integer {{math|''n'' ≥ 1}}, the action is {{visible anchor|n-transitive|text=''{{mvar|n}}-transitive''}} if {{math|''X''}} has at least {{math|''n''}} elements, and for any pair of {{math|''n''}}-tuples {{math|(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>), (''y''<sub>1</sub>, ..., ''y''<sub>''n''</sub>) ∈ ''X''<sup>''n''</sup>}} with pairwise distinct entries (that is {{math|''x''<sub>''i''</sub> ≠ ''x''<sub>''j''</sub>}}, {{math|''y''<sub>''i''</sub> ≠ ''y''<sub>''j''</sub>}} when {{math|''i'' ≠ ''j''}}) there exists a {{math|''g'' ∈ ''G''}} such that {{math|1=''g''&sdot;''x''<sub>''i''</sub> = ''y''<sub>''i''</sub>}} for {{math|1=''i'' = 1, ..., ''n''}}. In other words, the action on the subset of {{math|''X''<sup>''n''</sup>}} of tuples without repeated entries is transitive. For {{math|1=''n'' = 2, 3}} 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 {{mvar|n}}-transitive''}} when the action on tuples without repeated entries in {{math|''X''<sup>''n''</sup>}} is sharply transitive.

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

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

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

=== Topological properties ===
Assume that {{math|''X''}} is a [[topological space]] and the action of {{math|''G''}} is by [[homeomorphism]]s.

The action is ''wandering'' if every {{math|''x'' ∈ ''X''}} has a [[Neighbourhood (mathematics)|neighbourhood]] {{math|''U''}} such that there are only finitely many {{math|''g'' ∈ ''G''}} with {{math|''g''&sdot;''U'' ∩ ''U'' ≠ ∅}}.{{sfn|Thurston|1997|loc=Definition 3.5.1(iv)}}

More generally, a point {{math|''x'' ∈ ''X''}} is called a point of discontinuity for the action of {{math|''G''}} if there is an open subset {{math|''U'' ∋ ''x''}} such that there are only finitely many {{math|''g'' ∈ ''G''}} with {{math|''g''&sdot;''U'' ∩ ''U'' ≠ ∅}}. The ''domain of discontinuity'' of the action is the set of all points of discontinuity. Equivalently it is the largest {{math|''G''}}-stable open subset {{math|Ω ⊂ ''X''}} such that the action of {{math|''G''}} on {{math|Ω}} is wandering.{{sfn|Kapovich|2009|loc=p. 73}} In a dynamical context this is also called a ''[[wandering set]]''.

The action is ''properly discontinuous'' if for every [[Compact space|compact]] subset {{math|''K'' ⊂ ''X''}} there are only finitely many {{math|''g'' ∈ ''G''}} such that {{math|''g''&sdot;''K'' ∩ ''K'' ≠ ∅}}. This is strictly stronger than wandering; for instance the action of {{math|'''Z'''}} on {{math|'''R'''<sup>2</sup> &setminus; {{mset|(0, 0)}}}} given by {{math|1=''n''&sdot;(''x'', ''y'') = (2<sup>''n''</sup>''x'', 2<sup>−''n''</sup>''y'')}} is wandering and free but not properly discontinuous.{{sfn|Thurston|1980|p=176}}

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

An action of a group {{math|''G''}} on a [[locally compact space]] {{math|''X''}} is called ''[[Cocompact group action|cocompact]]'' if there exists a compact subset {{math|''A'' ⊂ ''X''}} such that {{math|1=''X'' = ''G'' &sdot; ''A''}}. For a properly discontinuous action, cocompactness is equivalent to compactness of the [[Quotient space (topology)|quotient space]] {{math|''X'' / ''G''}}.

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

The action is said to be ''{{visible anchor|proper}}'' if the map {{math|''G'' × ''X'' → ''X'' × ''X''}} defined by {{math|(''g'', ''x'') ↦ (''x'', ''g''&sdot;''x'')}} is [[proper map|proper]].{{sfn|tom Dieck|1987|loc=}} This means that given compact sets {{math|''K'', ''K''′}} the set of {{math|''g'' ∈ ''G''}} such that {{math|''g''&sdot;''K'' ∩ ''K''′ ≠ ∅}} is compact. In particular, this is equivalent to proper discontinuity if {{math|''G''}} is a [[discrete group]].

It is said to be ''locally free'' if there exists a neighbourhood {{math|''U''}} of {{math|''e''<sub>''G''</sub>}} such that {{math|''g''&sdot;''x'' ≠ ''x''}} for all {{math|''x'' ∈ ''X''}} and {{math|''g'' ∈ ''U'' &setminus; {{mset|''e''<sub>''G''</sub>}}}}.

The action is said to be ''strongly continuous'' if the orbital map {{math|''g'' ↦ ''g''&sdot;''x''}} is continuous for every {{math|''x'' ∈ ''X''}}. Contrary to what the name suggests, this is a weaker property than continuity of the action.{{citation needed|date=May 2023}}

If {{math|''G''}} is a [[Lie group]] and {{math|''X''}} a [[differentiable manifold]], then the subspace of ''smooth points'' for the action is the set of points {{math|''x'' ∈ ''X''}} such that the map {{math|''g'' ↦ ''g''&sdot;''x''}} is [[smooth map|smooth]]. There is a well-developed theory of [[Lie group action]]s, i.e. action which are smooth on the whole space.

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