Group Action (mathematics)
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Group Action Mathematics
Given an equilateral triangle, the counterclockwise rotation by 120° around the center of the triangle maps every vertex of the triangle to another one. The cyclic group C3 consisting of the rotations by 0°, 120° and 240° acts on the set of the three vertices.

In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group acts on the space or structure. If a group acts on a structure, it also acts on everything that is built on the 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 (finite-dimensional) vector space is called a representation of the group. It allows one to identify many groups with subgroups of GL(n, K), the group of the invertible matrices of dimension n over a field K.

The symmetric group Sn acts on any set with 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 G is a group with identity element e, and X is a set, then a (left) group action α of G on X is a function

(with α(g, x) often shortened to gx or gx when the action being considered is clear from context)

that satisfies the following two axioms:[1]

Identity:
Compatibility:

for all g and h in G and all x in X.

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

From these two axioms, it follows that for any fixed g in G, the function from X to itself which maps x to gx is a bijection, with inverse bijection the corresponding map for g−1. Therefore, one may equivalently define a group action of G on X as a group homomorphism from G into the symmetric group Sym(X) of all bijections from X to itself.[2]

Right group action

Likewise, a right group action of G on X is a function

(with α(x, g) often shortened to xg or xg when the action being considered is clear from context)

that satisfies the analogous axioms:

Identity:
Compatibility:

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 it is sufficient to only consider left actions without any loss of generality.

Types of actions

The action of G on X is called:

  • Transitive if X is non-empty and if for each pair x, y in X there exists a g in G such that . For example, the action of the symmetric group of X is transitive, the action of the general linear group or the special linear group of a vector space V on is transitive, but the action of the orthogonal group of a Euclidean space E is not transitive on (it is transitive on the unit sphere of E, though).
  • Faithful (or effective) if for every two distinct g, h in G there exists an x in X such that ; or equivalently, if for each in G there exists an x in X such that . In other words, in a faithful group action, different elements of G induce different permutations of X.[a] In algebraic terms, a group G acts faithfully on X if and only if the corresponding homomorphism to the symmetric group, , has a trivial kernel. Thus, for a faithful action, G embeds into a permutation group on X; specifically, G is isomorphic to its image in Sym(X). If G does not act faithfully on X, we can easily modify the group to obtain a faithful action. If we define , then N is a normal subgroup of G; indeed, it is the kernel of the homomorphism . The factor group G/N acts faithfully on X by setting . The original action of G on X is faithful if and only if . 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 S5, the icosahedral group, and the cyclic group . The smallest sets on which faithful actions can be defined are of size 5, 12, and 16 respectively.
    • The abelian groups of size 2n include a cyclic group as well as (the direct product of n copies of ), but the latter acts faithfully on a set of size 2n, whereas the former cannot act faithfully on a set smaller than itself.
  • Free (or semiregular or fixed point-free) if, given g, h in G, the existence of an x in X with implies . Equivalently: if g is a group element and there exists an x in X with (that is, if g has at least one fixed point), then g is the identity. Note that a free action on a non-empty set is faithful.
  • Regular (or simply transitive or sharply transitive) if it is both transitive and free; this is equivalent to saying that for every two x, y in X there exists precisely one g in G such that . In this case, X is called a principal homogeneous space for G or a G-torsor. The action of any group G on itself by left multiplication is regular, and thus faithful as well. Every group can, therefore, be embedded in the symmetric group on its own elements, Sym(G). This result is known as Cayley's theorem.
  • n-transitive if X has at least n elements, and for all distinct x1, ..., xn and all distinct y1, ..., yn, there is a g in G such that for . A 2-transitive action is also called doubly transitive, a 3-transitive action is also called triply transitive, and so on. Such actions define interesting classes of subgroups in the symmetric groups: 2-transitive groups and more generally multiply transitive groups. The action of the symmetric group on a set with n elements is always n-transitive; the action of the alternating group is (n-2)-transitive.
  • Sharply n-transitive if there is exactly one such g.
  • Primitive if it is transitive and preserves no non-trivial partition of X. See primitive permutation group for details.
  • Locally free if G is a topological group, and there is a neighbourhood U of e in G such that the restriction of the action to U is free; that is, if for some x and some g in U then .

Furthermore, if G acts on a topological space X, then the action is:

  • Wandering if every point x in X has a neighbourhood U such that is finite.[3] For example, the action of on by translations is wandering. The action of the modular group on the Poincaré half-plane is also wandering.
  • Properly discontinuous if X is a locally compact space and for every compact subset K ? X the set is finite. The wandering actions given above are also properly discontinuous. On the other hand, the action of on given by is wandering and free but not properly discontinuous.[4]
  • Proper if G is a topological group and the map from is proper.[5] If G is discrete then properness is equivalent to proper discontinuity for G-actions.
  • Said to have discrete orbits if the orbit of each x in X under the action of G is discrete in X.[3]
  • A covering space action if every point x in X has a neighbourhood U such that .[6]

If X is a non-zero module over a ring R and the action of G is R-linear then it is said to be

  • Irreducible if there is no nonzero proper invariant submodule.

Orbits and stabilizers

In the compound of five tetrahedra, the symmetry group is the (rotational) icosahedral group I of order 60, while the stabilizer of a single chosen tetrahedron is the (rotational) tetrahedral group T of order 12, and the orbit space I/T (of order 60/12 = 5) is naturally identified with the 5 tetrahedra - the coset gT corresponds to the tetrahedron to which g sends the chosen tetrahedron.

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:

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 if and only if there exists a g in G with . 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, .

The group action is transitive if and only if it has exactly one orbit, that is, if there exists x in X with . This is the case if and only if 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: 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, one writes GY for the set . The subset Y is said invariant under G if (which is equivalent to ). In that case, G also operates on Y by restricting the action to Y. The subset Y is called fixed under G if 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 such that for all . 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 , 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[7]) is the set of all elements in G that fix 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, , 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 . Then the two stabilizer groups Gx and Gy are related by . Proof: by definition, if and only if . Applying g-1 to both sides of this equality yields ; that is, . An opposite inclusion follows similarly by taking and supposing .

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 denote the conjugacy class of H. Then the orbit O has type if the stabilizer of some/any x in O belongs to . A maximal orbit type is often called a principal orbit type.

Orbit-stabilizer theorem and Burnside's lemma

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

.

In other words, g and h lie in the same coset for the stabilizer subgroup . Thus the fiber of f over any y in G·x is such a coset, and clearly every such coset occurs as a fiber. Therefore f defines a bijection between the set of cosets for the stabilizer subgroup and the orbit G·x, which sends .[8] This result is known as the orbit-stabilizer theorem.

If G is finite then the orbit-stabilizer theorem, together with Lagrange's theorem, gives

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 orbits of length 1 which are G-invariant elements.

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

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 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, . Applying the theorem now to the stabilizer G1, we can obtain . 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 which permutes 2,4,5 and 3,6,8, and fixes 1 and 7. Thus, . Applying the theorem a third time gives . 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 . One also sees that 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 .

A result closely related to the orbit-stabilizer theorem is Burnside's lemma:

where Xg 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 for all g in G and all x in X; that is, every group element induces the identity permutation on X.[9]
  • In every group G, left multiplication is an action of G on G: 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: 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: . An exponential notation is commonly used for the right-action variant: ; it satisfies (.
  • In every group G with subgroup H, conjugation is an action of G on conjugates of H: for all g in G and K conjugates of H.
  • The symmetric group Sn and its subgroups act on the set 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.
  • 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 and its subgroups, particularly its Lie subgroups (including the special linear group , orthogonal group , special orthogonal group , and symplectic group ) 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 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) transitive and free (that is, regular) action on these points;[10] indeed this can be used to give a definition of an affine space.
  • The projective linear group 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 , the symmetries of the projective line, which is sharply 3-transitive, preserving the cross ratio; the Möbius group 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 such that 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 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 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 acts on the set of real functions of a real variable in various ways, with (t?f)(x) equal to, for example, , , , , , or , but not .
  • Given a group action of G on X, we can define an induced action of G on the power set of X, by setting 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 , the mapping is a counterclockwise rotation through an angle ? about an axis given by a unit vector v; z is the same rotation; see quaternions and spatial rotation. Note that this is not a faithful action because the quaternion -1 leaves all points where they were, as does the quaternion 1.

Group actions and groupoids

The notion of group action can be put in a broader context by using the action groupoid associated to the group action, thus allowing techniques from groupoid theory such as presentations and fibrations. Further the stabilizers of the action are the vertex groups, and the orbits of the action are the components, of the action groupoid. For more details, see the book Topology and groupoids referenced below.

This action groupoid comes with a morphism p: G? -> G which is a covering morphism of groupoids. This allows a relation between such morphisms and covering maps in topology.

Morphisms and isomorphisms between G-sets

If X and Y are two G-sets, a morphism from X to Y is a function such that 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 , where S is some set and G acts on 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).

Continuous group actions

One often considers continuous group actions: the group G is a topological group, X is a topological space, and the map is continuous with respect to the product topology of . The space X is also called a G-space in this case. This is indeed a generalization, since every group can be considered a topological group by using the discrete topology. All the concepts introduced above still work in this context, however we define morphisms between G-spaces to be continuous maps compatible with the action of G. The quotient X/G inherits the quotient topology from X, and is called the quotient space of the action. The above statements about isomorphisms for regular, free and transitive actions are no longer valid for continuous group actions.

If X is a regular covering space of another topological space Y, then the action of the deck transformation group on X is properly discontinuous as well as being free. Every free, properly discontinuous action of a group G on a path-connected topological space X arises in this manner: the quotient map is a regular covering map, and the deck transformation group is the given action of G on X. Furthermore, if X is simply connected, the fundamental group of X/G will be isomorphic to G.

These results have been generalized in the book Topology and Groupoids referenced below to obtain the fundamental groupoid of the orbit space of a discontinuous action of a discrete group on a Hausdorff space, as, under reasonable local conditions, the orbit groupoid of the fundamental groupoid of the space. This allows calculations such as the fundamental group of the symmetric square of a space X, namely the orbit space of the product of X with itself under the twist action of the cyclic group of order 2 sending to .

An action of a group G on a locally compact space X is cocompact if there exists a compact subset A of X such that . For a properly discontinuous action, cocompactness is equivalent to compactness of the quotient space X/G.

The action of G on X is said to be proper if the mapping that sends is a proper map.

Strongly continuous group action and smooth points

A group action of a topological group G on a topological space X is said to be strongly continuous if for all x in X, the map is continuous with respect to the respective topologies. Such an action induces an action on the space of continuous functions on X by defining for every g in G, f a continuous function on X, and x in X. Note that, while every continuous group action is strongly continuous, the converse is not in general true.[11]

The subspace of smooth points for the action is the subspace of X of points x such that is smooth, that is, it is continuous and all derivatives[where?] are continuous.

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. 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. A morphism between G-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

  1. ^ that is, the associated permutation representation is injective.

Citations

  1. ^ Eie & Chang (2010). A Course on Abstract Algebra. p. 144.
  2. ^ This is done, for example, by Smith (2008). Introduction to abstract algebra. p. 253.
  3. ^ a b Thurston, William (1980), The geometry and topology of three-manifolds, Princeton lecture notes, p. 175
  4. ^ Thurston 1980, p. 176.
  5. ^ 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
  6. ^ Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. p. 72. ISBN 0-521-79540-0.
  7. ^ Procesi, Claudio (2007). Lie Groups: An Approach through Invariants and Representations. Springer Science & Business Media. p. 5. ISBN 9780387289298. Retrieved 2017.
  8. ^ M. Artin, Algebra, Proposition 6.4 on p. 179
  9. ^ Eie & Chang (2010). A Course on Abstract Algebra. p. 145.
  10. ^ Reid, Miles (2005). Geometry and topology. Cambridge, UK New York: Cambridge University Press. p. 170. ISBN 9780521613255.
  11. ^ Yuan, Qiaochu (27 February 2013). "wiki's definition of "strongly continuous group action" wrong?". Mathematics Stack Exchange. Retrieved 2013.

References

External links


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