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Group action

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File:Group action on equilateral triangle.svg
The cyclic group Template:Math consisting of the rotations by 0°, 120° and 240° acts on the set of the three vertices.

In mathematics, a group action of a group <math>G</math> on a set <math>S</math> is a group homomorphism from <math>G</math> to some group (under function composition) of functions from <math>S</math> to itself. It is said that <math>G</math> acts on <math>S</math>.

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 <math>\operatorname{GL}(n,K)</math>, the group of the invertible matrices of dimension <math>n</math> over a field <math>K</math>.

The symmetric group <math>S_n</math> acts on any set with <math>n</math> 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.

DefinitionEdit

Left group actionEdit

If <math>G</math> is a 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 Template:Mvar is a function

<math>\alpha : G \times X \to X</math>

that satisfies the following two axioms:<ref>Template:Cite book</ref>

Identity: <math>\alpha(e,x)=x</math>
Compatibility: <math>\alpha(g,\alpha(h,x))=\alpha(gh,x)</math>

for all Template:Mvar and Template:Mvar in Template:Mvar and all Template:Mvar in <math>X</math>.

The group <math>G</math> is then said to act on <math>X</math> (from the left). A set <math>X</math> together with an action of <math>G</math> is called a (left) <math>G</math>-set.

It can be notationally convenient to curry the action <math>\alpha</math>, so that, instead, one has a collection of transformations Template:Math, with one transformation Template:Math for each group element Template:Math. The identity and compatibility relations then read

<math>\alpha_e(x) = x</math>

and

<math>\alpha_g(\alpha_h(x)) = (\alpha_g \circ \alpha_h)(x) = \alpha_{gh}(x)</math>

The second axiom states that the function composition is compatible with the group multiplication; they form a commutative diagram. This axiom can be shortened even further, and written as <math>\alpha_g\circ\alpha_h=\alpha_{gh}</math>.

With the above understanding, it is very common to avoid writing <math>\alpha</math> entirely, and to replace it with either a dot, or with nothing at all. Thus, Template:Math can be shortened to Template:Math or Template:Math, especially when the action is clear from context. The axioms are then

<math>e{\cdot}x = x</math>
<math>g{\cdot}(h{\cdot}x) = (gh){\cdot}x</math>

From these two axioms, it follows that for any fixed Template:Mvar in <math>G</math>, the function from Template:Mvar to itself which maps Template:Mvar to Template:Math is a bijection, with inverse bijection the corresponding map for Template:Math. Therefore, one may equivalently define a group action of Template:Mvar on Template:Mvar as a group homomorphism from Template:Mvar into the symmetric group Template:Math of all bijections from Template:Mvar to itself.<ref>This is done, for example, by Template:Cite book</ref>

Right group actionEdit

Likewise, a right group action of <math>G</math> on <math>X</math> is a function

<math>\alpha : X \times G \to X,</math>

that satisfies the analogous axioms:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Identity: <math>\alpha(x,e)=x</math>
Compatibility: <math>\alpha(\alpha(x,g),h)=\alpha(x,gh)</math>

(with Template:Math often shortened to Template:Math or Template:Math when the action being considered is clear from context)

Identity: <math>x{\cdot}e = x</math>
Compatibility: <math>(x{\cdot}g){\cdot}h = x{\cdot}(gh)</math>

for all Template:Mvar and Template:Mvar in Template:Mvar and all Template:Mvar in Template:Mvar.

The difference between left and right actions is in the order in which a product Template:Math acts on Template:Mvar. For a left action, Template:Mvar acts first, followed by Template:Mvar second. For a right action, Template:Mvar acts first, followed by Template:Mvar second. Because of the formula Template:Math, 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 Template:Mvar on Template:Mvar can be considered as a left action of its opposite group Template:Math on Template:Mvar.

Thus, for establishing general properties of group actions, it suffices to consider only left actions. However, there are cases where this is not possible. For example, the multiplication of a group induces both a left action and a right action on the group itself—multiplication on the left and on the right, respectively.

Notable properties of actionsEdit

Let Template:Math be a group acting on a set Template:Math. The action is called Template:Visible anchor or Template:Visible anchor if Template:Math for all Template:Math implies that Template:Math. Equivalently, the homomorphism from Template:Math to the group of bijections of Template:Math corresponding to the action is injective.

The action is called Template:Visible anchor (or semiregular or fixed-point free) if the statement that Template:Math for some Template:Math already implies that Template:Math. In other words, no non-trivial element of Template:Math fixes a point of Template: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 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 Template:Math (of cardinality Template:Math) acts faithfully on a set of size Template:Math. This is not always the case, for example the cyclic group Template:Math cannot act faithfully on a set of size less than Template: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 Template:Math, the icosahedral group Template:Math and the cyclic group Template:Math. The smallest sets on which faithful actions can be defined for these groups are of size 5, 7, and 16 respectively.

Transitivity propertiesEdit

The action of Template:Math on Template:Math is called Template:Visible anchor if for any two points Template:Math there exists a Template:Math so that Template:Math.

The action is Template:Visible anchor (or sharply transitive, or Template:Visible anchor) if it is both transitive and free. This means that given Template:Math there is exactly one Template:Math such that Template:Math. If Template:Math is acted upon simply transitively by a group Template:Math then it is called a principal homogeneous space for Template:Math or a Template:Math-torsor.

For an integer Template:Math, the action is Template:Visible anchor if Template:Math has at least Template:Math elements, and for any pair of Template:Math-tuples Template:Math with pairwise distinct entries (that is Template:Math, Template:Math when Template:Math) there exists a Template:Math such that Template:Math for Template:Math. In other words, the action on the subset of Template:Math of tuples without repeated entries is transitive. For Template:Math 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 Template:Visible anchor when the action on tuples without repeated entries in Template:Math is sharply transitive.

ExamplesEdit

The action of the symmetric group of Template:Math is transitive, in fact Template:Math-transitive for any Template:Math up to the cardinality of Template:Math. If Template:Math has cardinality Template:Math, the action of the alternating group is Template:Math-transitive but not Template:Math-transitive.

The action of the general linear group of a vector space Template:Math on the set Template:Math of non-zero vectors is transitive, but not 2-transitive (similarly for the action of the special linear group if the dimension of Template:Math 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 actionsEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The action of Template:Math on Template:Math is called primitive if there is no partition of Template:Math preserved by all elements of Template:Math apart from the trivial partitions (the partition in a single piece and its dual, the partition into singletons).

Topological propertiesEdit

Assume that Template:Math is a topological space and the action of Template:Math is by homeomorphisms.

The action is wandering if every Template:Math has a neighbourhood Template:Math such that there are only finitely many Template:Math with Template:Math.Template:Sfn

More generally, a point Template:Math is called a point of discontinuity for the action of Template:Math if there is an open subset Template:Math such that there are only finitely many Template:Math with Template:Math. The domain of discontinuity of the action is the set of all points of discontinuity. Equivalently it is the largest Template:Math-stable open subset Template:Math such that the action of Template:Math on Template:Math is wandering.Template:Sfn In a dynamical context this is also called a wandering set.

The action is properly discontinuous if for every compact subset Template:Math there are only finitely many Template:Math such that Template:Math. This is strictly stronger than wandering; for instance the action of Template:Math on Template:Math given by Template:Math is wandering and free but not properly discontinuous.Template:Sfn

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 Template:Math has a neighbourhood Template:Math such that Template:Math for every Template:Math.Template:Sfn 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.Template:Sfn

An action of a group Template:Math on a locally compact space Template:Math is called cocompact if there exists a compact subset Template:Math such that Template:Math. For a properly discontinuous action, cocompactness is equivalent to compactness of the quotient space Template:Math.

Actions of topological groupsEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Now assume Template:Math is a topological group and Template:Math a topological space on which it acts by homeomorphisms. The action is said to be continuous if the map Template:Math is continuous for the product topology.

The action is said to be Template:Visible anchor if the map Template:Math defined by Template:Math is proper.Template:Sfn This means that given compact sets Template:Math the set of Template:Math such that Template:Math is compact. In particular, this is equivalent to proper discontinuity if Template:Math is a discrete group.

It is said to be locally free if there exists a neighbourhood Template:Math of Template:Math such that Template:Math for all Template:Math and Template:Math.

The action is said to be strongly continuous if the orbital map Template:Math is continuous for every Template:Math. Contrary to what the name suggests, this is a weaker property than continuity of the action.Template:Citation needed

If Template:Math is a Lie group and Template:Math a differentiable manifold, then the subspace of smooth points for the action is the set of points Template:Math such that the map Template:Math is smooth. There is a well-developed theory of Lie group actions, i.e. action which are smooth on the whole space.

Linear actionsEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} If Template:Math acts by linear transformations on a module over a commutative ring, the action is said to be irreducible if there are no proper nonzero Template:Math-invariant submodules. It is said to be semisimple if it decomposes as a direct sum of irreducible actions.

Orbits and stabilizersEdit

File:Compound of five tetrahedra.png
In the compound of five tetrahedra, the symmetry group is the (rotational) icosahedral group Template:Math of order 60, while the stabilizer of a single chosen tetrahedron is the (rotational) tetrahedral group Template:Math of order 12, and the orbit space Template:Math (of order 60/12 = 5) is naturally identified with the 5 tetrahedra – the coset Template:Math corresponds to the tetrahedron to which Template:Math sends the chosen tetrahedron.

Consider a group Template:Math acting on a set Template:Math. The Template:Visible anchor of an element Template:Math in Template:Math is the set of elements in Template:Math to which Template:Math can be moved by the elements of Template:Math. The orbit of Template:Math is denoted by Template:Math: <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 Template:Math in) Template:Math under the action of Template:Math form a partition of Template:Math. The associated equivalence relation is defined by saying Template:Math if and only if there exists a Template:Math in Template:Math with Template:Math. The orbits are then the equivalence classes under this relation; two elements Template:Math and Template:Math are equivalent if and only if their orbits are the same, that is, Template:Math.

The group action is transitive if and only if it has exactly one orbit, that is, if there exists Template:Math in Template:Math with Template:Math. This is the case if and only if Template:Math for Template:Em Template:Math in Template:Math (given that Template:Math is non-empty).

The set of all orbits of Template:Math under the action of Template:Math is written as Template:Math (or, less frequently, as Template:Math), and is called the Template:Visible anchor of the action. In geometric situations it may be called the Template:Visible anchor, while in algebraic situations it may be called the space of Template:Visible anchor, and written Template:Math, by contrast with the invariants (fixed points), denoted Template:Math: the coinvariants are a Template:Em while the invariants are a Template:Em. The coinvariant terminology and notation are used particularly in group cohomology and group homology, which use the same superscript/subscript convention.

Invariant subsetsEdit

If Template:Math is a subset of Template:Math, then Template:Math denotes the set Template:Math. The subset Template:Math is said to be invariant under Template:Math if Template:Math (which is equivalent Template:Math). In that case, Template:Math also operates on Template:Math by restricting the action to Template:Math. The subset Template:Math is called fixed under Template:Math if Template:Math for all Template:Math in Template:Math and all Template:Math in Template:Math. Every subset that is fixed under Template:Math is also invariant under Template:Math, but not conversely.

Every orbit is an invariant subset of Template:Math on which Template:Math acts transitively. Conversely, any invariant subset of Template:Math is a union of orbits. The action of Template:Math on Template:Math is transitive if and only if all elements are equivalent, meaning that there is only one orbit.

A Template:Math-invariant element of Template:Math is Template:Math such that Template:Math for all Template:Math. The set of all such Template:Math is denoted Template:Math and called the Template:Math-invariants of Template:Math. When Template:Math is a [[G-module|Template:Math-module]], Template:Math is the zeroth cohomology group of Template:Math with coefficients in Template:Math, and the higher cohomology groups are the derived functors of the functor of Template:Math-invariants.

Fixed points and stabilizer subgroupsEdit

Given Template:Math in Template:Math and Template:Math in Template:Math with Template:Math, it is said that "Template:Math is a fixed point of Template:Math" or that "Template:Math fixes Template:Math". For every Template:Math in Template:Math, the Template:Visible anchor of Template:Math with respect to Template:Math (also called the isotropy group or little group<ref name="Procesi">Template:Cite book</ref>) is the set of all elements in Template:Math that fix Template:Math: <math display=block>G_x = \{g \in G : g{\cdot}x = x\}.</math> This is a subgroup of Template:Math, though typically not a normal one. The action of Template:Math on Template:Math is free if and only if all stabilizers are trivial. The kernel Template:Math of the homomorphism with the symmetric group, Template:Math, is given by the intersection of the stabilizers Template:Math for all Template:Math in Template:Math. If Template:Math is trivial, the action is said to be faithful (or effective).

Let Template:Math and Template:Math be two elements in Template:Math, and let Template:Math be a group element such that Template:Math. Then the two stabilizer groups Template:Math and Template:Math are related by Template:Math. Proof: by definition, Template:Math if and only if Template:Math. Applying Template:Math to both sides of this equality yields Template:Math; that is, Template:Math. An opposite inclusion follows similarly by taking Template:Math and Template:Math.

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 Template:Math (that is, the set of all conjugates of the subgroup). Let Template:Math denote the conjugacy class of Template:Math. Then the orbit Template:Math has type Template:Math if the stabilizer Template:Math of some/any Template:Math in Template:Math belongs to Template:Math. A maximal orbit type is often called a principal orbit type.

Template:Visible anchorEdit

Orbits and stabilizers are closely related. For a fixed Template:Math in Template:Math, consider the map Template:Math given by Template:Math. By definition the image Template:Math of this map is the orbit Template:Math. 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, Template:Math if and only if Template:Math and Template:Math lie in the same coset for the stabilizer subgroup Template:Math. Thus, the fiber Template:Math of Template:Math over any Template:Math in Template:Math is contained in such a coset, and every such coset also occurs as a fiber. Therefore Template:Math induces a Template:Em between the set Template:Math of cosets for the stabilizer subgroup and the orbit Template:Math, which sends Template:Math.<ref>M. Artin, Algebra, Proposition 6.8.4 on p. 179</ref> This result is known as the orbit-stabilizer theorem.

If Template:Math is finite then the orbit-stabilizer theorem, together with 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 Template:Math 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 Template:Math be a group of prime order Template:Math acting on a set Template:Math with Template:Math elements. Since each orbit has either Template:Math or Template:Math elements, there are at least Template:Math orbits of length Template:Math which are Template:Math-invariant elements. More specifically, Template:Math and the number of Template:Math-invariant elements are congruent modulo Template:Math.<ref>Template:Cite book</ref>

This result is especially useful since it can be employed for counting arguments (typically in situations where Template:Math is finite as well).

File:Labeled cube graph.png
Cubical graph with vertices labeled
Example: We can use the orbit-stabilizer theorem to count the automorphisms of a graph. Consider the cubical graph as pictured, and let Template:Math denote its automorphism group. Then Template:Math acts on the set of vertices Template:Math, and this action is transitive as can be seen by composing rotations about the center of the cube. Thus, by the orbit-stabilizer theorem, Template:Math. Applying the theorem now to the stabilizer Template:Math, we can obtain Template:Math. Any element of Template:Math 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 Template:Math, which permutes 2, 4, 5 and 3, 6, 8, and fixes 1 and 7. Thus, Template:Math. Applying the theorem a third time gives Template:Math. Any element of Template:Math 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 Template:Math. One also sees that Template:Math consists only of the identity automorphism, as any element of Template:Math 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 Template:Math.

Burnside's lemmaEdit

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 Template:Math is the set of points fixed by Template:Math. This result is mainly of use when Template:Math and Template:Math 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 Template:Math, the set of formal differences of finite Template:Math-sets forms a ring called the Burnside ring of Template:Math, where addition corresponds to disjoint union, and multiplication to Cartesian product.

ExamplesEdit

Template:Fix }}

Group actions and groupoidsEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The notion of group action can be encoded by the action groupoid Template:Math 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-setsEdit

If Template:Math and Template:Math are two Template:Math-sets, a morphism from Template:Math to Template:Math is a function Template:Math such that Template:Math for all Template:Math in Template:Math and all Template:Math in Template:Math. Morphisms of Template:Math-sets are also called equivariant maps or Template:Math-maps.

The composition of two morphisms is again a morphism. If a morphism Template:Math is bijective, then its inverse is also a morphism. In this case Template:Math is called an isomorphism, and the two Template:Math-sets Template:Math and Template:Math are called isomorphic; for all practical purposes, isomorphic Template:Math-sets are indistinguishable.

Some example isomorphisms:

With this notion of morphism, the collection of all Template:Math-sets forms a category; this category is a Grothendieck topos (in fact, assuming a classical metalogic, this topos will even be Boolean).

Variants and generalizationsEdit

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 Template:Math of some category, and then define an action on Template:Math as a monoid homomorphism into the monoid of endomorphisms of Template:Math. If Template:Math 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 Template:Math as a category with a single object in which every morphism is invertible.<ref>Template:Harvp</ref> A (left) group action is then nothing but a (covariant) functor from Template:Math to the category of sets, and a group representation is a functor from Template:Math to the category of vector spaces.<ref>Template:Harvp</ref> A morphism between Template:Math-sets is then a natural transformation between the group action functors.<ref>Template:Harvp</ref> In analogy, an action of a groupoid is a functor from the groupoid to the category of sets or to some other category.

In addition to continuous 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.

GalleryEdit

  • Octahedral-group-action.png

    Orbit of a fundamental spherical triangle (marked in red) under action of the full octahedral group.

  • Icosahedral-group-action.png

    Orbit of a fundamental spherical triangle (marked in red) under action of the full icosahedral group.

See alsoEdit

NotesEdit

Template:Notelist

CitationsEdit

Template:Reflist

ReferencesEdit

External linksEdit

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:GroupAction%7CGroupAction.html}} |title = Group Action |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}

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