Editing Group action (section)

== <span id="orbstab"></span><span id="quotient"></span> Orbits and stabilizers ==
<!-- This section is linked from [[Symmetry]] -->
[[File:Compound of five tetrahedra.png|thumb|In the [[compound of five tetrahedra]], the symmetry group is the (rotational) icosahedral group {{math|''I''}} of order 60, while the stabilizer of a single chosen tetrahedron is the (rotational) [[tetrahedral group]] {{math|''T''}} of order 12, and the orbit space {{math|''I'' / ''T''}} (of order 60/12&nbsp;=&nbsp;5) is naturally identified with the 5 tetrahedra – the coset {{math|''gT''}} corresponds to the tetrahedron to which {{math|''g''}} sends the chosen tetrahedron.]]

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

The defining properties of a group guarantee that the set of orbits of (points {{math|''x''}} in) {{math|''X''}} under the action of {{math|''G''}} form a [[Partition of a set|partition]] of {{math|''X''}}. The associated [[equivalence relation]] is defined by saying {{math|''x'' ~ ''y''}} [[if and only if]] there exists a {{math|''g''}} in {{math|''G''}} with {{math|1=''g''&sdot;''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''&sdot;''x'' = ''G''&sdot;''y''}}.

The group action is [[Group action (mathematics)#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''&sdot;''x'' = ''X''}}. This is the case if and only if {{math|1=''G''&sdot;''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.

=== Invariant subsets ===
If {{math|''Y''}} is a [[subset]] of {{math|''X''}}, then {{math|''G''&sdot;''Y''}} denotes the set {{math|{{mset|''g''&sdot;''y'' : ''g'' ∈ ''G'' and ''y'' ∈ ''Y''}}}}. The subset {{math|''Y''}} is said to be ''invariant under ''{{math|''G''}} if {{math|1=''G''&sdot;''Y'' = ''Y''}} (which is equivalent {{math|''G''&sdot;''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''&sdot;''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 [[Group action (mathematics)#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''&sdot;''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.

=== Fixed points and stabilizer subgroups ===
Given {{math|''g''}} in {{math|''G''}} and {{math|''x''}} in {{math|''X''}} with {{math|1=''g''&sdot;''x'' = ''x''}}, it is said that "{{math|''x''}} is a fixed point of {{math|''g''}}" or that "{{math|''g''}} fixes {{math|''x''}}". For every {{math|''x''}} in {{math|''X''}}, the '''{{visible anchor|stabilizer subgroup}}''' of {{math|''G''}} with respect to {{math|''x''}} (also called the '''isotropy group''' or '''little group'''<ref name="Procesi">{{cite book|last1=Procesi|first1=Claudio|title=Lie Groups: An Approach through Invariants and Representations|date=2007|publisher=Springer Science & Business Media|isbn=9780387289298|page=5|url=https://books.google.com/books?id=Sl8OAGYRz_AC&q=%22little+group%22+action&pg=PA5|access-date=23 February 2017|language=en}}</ref>) is the set of all elements in {{math|''G''}} that fix {{math|''x''}}:
<math display=block>G_x = \{g \in G : g{\cdot}x = x\}.</math>
This is a [[subgroup]] of {{math|''G''}}, though typically not a normal one. The action of {{math|''G''}} on {{math|''X''}} is [[Group action (mathematics)#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''&sdot;''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>}}. Proof: by definition, {{math|''h'' ∈ ''G''<sub>''y''</sub>}} if and only if {{math|1=''h''&sdot;(''g''&sdot;''x'') = ''g''&sdot;''x''}}. Applying {{math|''g''<sup>−1</sup>}} to both sides of this equality yields {{math|1=(''g''<sup>−1</sup>''hg'')&sdot;''x'' = ''x''}}; that is, {{math|''g''<sup>−1</sup>''hg'' ∈ ''G''<sub>''x''</sub>}}. An opposite inclusion follows similarly by taking {{math|''h'' ∈ ''G''<sub>''x''</sub>}} and {{math|1=''x'' = ''g''<sup>−1</sup>&sdot;''y''}}.

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

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

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

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

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

[[File:Labeled cube graph.png|thumb|Cubical graph with vertices labeled]]
: '''Example:'''  We can use the orbit-stabilizer theorem to count the automorphisms of a [[Graph (discrete mathematics)|graph]]. Consider the [[cubical graph]] as pictured, and let {{math|''G''}} denote its [[Graph automorphism|automorphism]] group. Then {{math|''G''}} acts on the set of vertices {{math|{{mset|1, 2, ..., 8}}}}, and this action is transitive as can be seen by composing rotations about the center of the cube. Thus, by the orbit-stabilizer theorem, {{math|1={{abs|''G''}} = {{abs|''G'' &sdot; 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>) &sdot; 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''&pi;''/3}}, which permutes 2, 4, 5 and 3, 6, 8, and fixes 1 and 7. Thus, {{math|1={{abs|(''G''<sub>1</sub>) &sdot; 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>) &sdot; 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>) &sdot; 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 &sdot; 3 &sdot; 2 &sdot; 1 = 48}}.

=== Burnside's lemma ===
A result closely related to the orbit-stabilizer theorem is [[Burnside's lemma]]:
<math display=block>|X/G|=\frac{1}{|G|}\sum_{g\in G} |X^g|,</math>
where {{math|''X''<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.

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