Английская Википедия:Group action

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

In mathematics, 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 acts also on triangles by transforming triangles into triangles.

Formally, a group action of a group Шаблон:Math on a set Шаблон:Math is a group homomorphism from Шаблон:Math to some group (under function composition) of functions from Шаблон:Math to itself.

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 Шаблон:Math, the group of the invertible matrices of dimension Шаблон:Math over a field Шаблон:Math.

The symmetric group Шаблон:Math acts on any set with Шаблон: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.

Definition

Left group action

If Шаблон:Mvar is a group with identity element Шаблон:Mvar, and Шаблон:Mvar is a set, then a (left) group action Шаблон:Mvar of Шаблон:Mvar on Шаблон:Mvar is a function

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

that satisfies the following two axioms:[1]

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

for all Шаблон:Mvar and Шаблон:Mvar in Шаблон:Mvar and all Шаблон:Mvar in Шаблон:Mvar.

The group Шаблон:Mvar is then said to act on Шаблон:Mvar (from the left). A set Шаблон:Mvar together with an action of Шаблон:Mvar is called a (left) Шаблон:Mvar-set.

It can be notationally convenient to curry the action Шаблон:Math, so that, instead, one has a collection of transformations Шаблон:Math, with one transformation Шаблон:Math for each group element Шаблон: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>

with Шаблон:Math being function composition. The second axiom then 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 Шаблон:Math.

With the above understanding, it is very common to avoid writing Шаблон:Math entirely, and to replace it with either a dot, or with nothing at all. Thus, Шаблон:Math can be shortened to Шаблон:Math or Шаблон: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 Шаблон:Mvar in Шаблон:Mvar, the function from Шаблон:Mvar to itself which maps Шаблон:Mvar to Шаблон:Math is a bijection, with inverse bijection the corresponding map for Шаблон:Math. Therefore, one may equivalently define a group action of Шаблон:Mvar on Шаблон:Mvar as a group homomorphism from Шаблон:Mvar into the symmetric group Шаблон:Math of all bijections from Шаблон:Mvar to itself.[2]

Right group action

Likewise, a right group action of Шаблон:Mvar on Шаблон:Mvar is a function

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

that satisfies the analogous axioms:[3]

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

(with Шаблон:Math often shortened to Шаблон:Math or Шаблон: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 Шаблон:Mvar and Шаблон:Mvar in Шаблон:Mvar and all Шаблон:Mvar in Шаблон:Mvar.

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

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

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

Transitivity properties

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

The action is Шаблон:Visible anchor (or sharply transitive, or Шаблон:Visible anchor) if it is both transitive and free. This means that given Шаблон:Math the element Шаблон:Math in the definition of transitivity is unique. If Шаблон:Math is acted upon simply transitively by a group Шаблон:Math then it is called a principal homogeneous space for Шаблон:Math or a Шаблон:Math-torsor.

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

Examples

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

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

Шаблон:Main The action of Шаблон:Math on Шаблон:Math is called primitive if there is no partition of Шаблон:Math preserved by all elements of Шаблон:Math apart from the trivial partitions (the partition in a single piece and its dual, the partition into singletons).

Topological properties

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

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

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

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

The action by deck transformations of the fundamental group of a locally simply connected space on an covering space is wandering and free. Such actions can be characterized by the following property: every Шаблон:Math has a neighbourhood Шаблон:Math such that Шаблон:Math for every Шаблон:Math.Шаблон: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.Шаблон:Sfn

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

Actions of topological groups

Шаблон:Main Now assume Шаблон:Math is a topological group and Шаблон:Math a topological space on which it acts by homeomorphisms. The action is said to be continuous if the map Шаблон:Math is continuous for the product topology.

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

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

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

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

Linear actions

Шаблон:Main If Шаблон: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 Шаблон:Math-invariant submodules. It is said to be semisimple if it decomposes as a direct sum of irreducible actions.

Orbits and stabilizers

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

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

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

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

Invariant subsets

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

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

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

Fixed points and stabilizer subgroups

Given Шаблон:Math in Шаблон:Math and Шаблон:Math in Шаблон:Math with Шаблон:Math, it is said that "Шаблон:Math is a fixed point of Шаблон:Math" or that "Шаблон:Math fixes Шаблон:Math". For every Шаблон:Math in Шаблон:Math, the Шаблон:Visible anchor of Шаблон:Math with respect to Шаблон:Math (also called the isotropy group or little group[4]) is the set of all elements in Шаблон:Math that fix Шаблон:Math: <math display=block>G_x = \{g \in G : g{\cdot}x = x\}.</math> This is a subgroup of Шаблон:Math, though typically not a normal one. The action of Шаблон:Math on Шаблон:Math is free if and only if all stabilizers are trivial. The kernel Шаблон:Math of the homomorphism with the symmetric group, Шаблон:Math, is given by the intersection of the stabilizers Шаблон:Math for all Шаблон:Math in Шаблон:Math. If Шаблон:Math is trivial, the action is said to be faithful (or effective).

Let Шаблон:Math and Шаблон:Math be two elements in Шаблон:Math, and let Шаблон:Math be a group element such that Шаблон:Math. Then the two stabilizer groups Шаблон:Math and Шаблон:Math are related by Шаблон:Math. Proof: by definition, Шаблон:Math if and only if Шаблон:Math. Applying Шаблон:Math to both sides of this equality yields Шаблон:Math; that is, Шаблон:Math. An opposite inclusion follows similarly by taking Шаблон:Math and Шаблон: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 Шаблон:Math (that is, the set of all conjugates of the subgroup). Let Шаблон:Math denote the conjugacy class of Шаблон:Math. Then the orbit Шаблон:Math has type Шаблон:Math if the stabilizer Шаблон:Math of some/any Шаблон:Math in Шаблон:Math belongs to Шаблон:Math. A maximal orbit type is often called a principal orbit type.

Шаблон:Visible anchor and Burnside's lemma

Orbits and stabilizers are closely related. For a fixed Шаблон:Math in Шаблон:Math, consider the map Шаблон:Math given by Шаблон:Math. By definition the image Шаблон:Math of this map is the orbit Шаблон: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, Шаблон:Math if and only if Шаблон:Math and Шаблон:Math lie in the same coset for the stabilizer subgroup Шаблон:Math. Thus, the fiber Шаблон:Math of Шаблон:Math over any Шаблон:Math in Шаблон:Math is contained in such a coset, and every such coset also occurs as a fiber. Therefore Шаблон:Math induces a Шаблон:Em between the set Шаблон:Math of cosets for the stabilizer subgroup and the orbit Шаблон:Math, which sends Шаблон:Math.[5] This result is known as the orbit-stabilizer theorem.

If Шаблон: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 Шаблон: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 Шаблон:Math be a group of prime order Шаблон:Math acting on a set Шаблон:Math with Шаблон:Math elements. Since each orbit has either Шаблон:Math or Шаблон:Math elements, there are at Шаблон:Math orbits of length Шаблон:Math which are Шаблон:Math-invariant elements.

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

Файл: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 Шаблон:Math denote its automorphism group. Then Шаблон:Math acts on the set of vertices Шаблон: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, Шаблон:Math. Applying the theorem now to the stabilizer Шаблон:Math, we can obtain Шаблон:Math. Any element of Шаблон: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 Шаблон:Math, which permutes 2, 4, 5 and 3, 6, 8, and fixes 1 and 7. Thus, Шаблон:Math. Applying the theorem a third time gives Шаблон:Math. Any element of Шаблон: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 Шаблон:Math. One also sees that Шаблон:Math consists only of the identity automorphism, as any element of Шаблон: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 Шаблон:Math.

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

Examples

Group actions and groupoids

Шаблон:Main The notion of group action can be encoded by the action groupoid Шаблон: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-sets

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

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

Some example isomorphisms:

With this notion of morphism, the collection of all Шаблон: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 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 Шаблон:Math of some category, and then define an action on Шаблон:Math as a monoid homomorphism into the monoid of endomorphisms of Шаблон:Math. If Шаблон: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 Шаблон:Math as a category with a single object in which every morphism is invertible. A (left) group action is then nothing but a (covariant) functor from Шаблон:Math to the category of sets, and a group representation is a functor from Шаблон:Math to the category of vector spaces. A morphism between Шаблон:Math-sets is then a natural transformation between the group action functors. 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

See also

Notes

Шаблон:Notelist

Citations

Шаблон:Reflist

References

External links

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  1. ↑ Шаблон:Cite book
  2. ↑ This is done, for example, by Шаблон:Cite book
  3. ↑ Шаблон:Cite web
  4. ↑ Шаблон:Cite book
  5. ↑ M. Artin, Algebra, Proposition 6.8.4 on p. 179
  6. ↑ Шаблон:Cite book
  7. ↑ Шаблон:Cite book
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