In mathematics, a symmetry group describes all symmetries of objects. This is formalized by the notion of a group action: every element of the group "acts" like a bijective map (or "symmetry") on some set. In this case, the group is also called a permutation group (especially if the set is finite or not a vector space) or transformation group (especially if the set is a vector space and the group acts like linear transformations of the set). A permutation representation of a group G is a representation of G as a group of permutations of the set (usually if the set is finite), and may be described as a group representation of G by permutation matrices, and is usually considered in the finite-dimensional case—it is the same as a group action of G on an ordered basis of a vector space. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... The symmetry group of an object (e. ... Sphere symmetry group o. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself); the relationship is often written as (G,M). ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In mathematics, especially in abstract algebra and related areas, a permutation is a bijection from a finite set X onto itself. ... Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In linear algebra, a permutation matrix is a binary matrix that has exactly one entry 1 in each row and each column and 0s elsewhere. ... In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...

Definition

If G is a group and X is a set, then a (left) group action of G on X is a binary function In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, a binary function, or function of two variables, is like a function, except that it has two inputs instead of one. ...

denoted

which satisfies the following two axioms:

1. (gh)·x = g·(h·x) for all g, h in G and x in X
2. e·x = x for every x in X (where e denotes the identity element of G)

The set X is called a (left) G-set. The group G is said to act on X (on the left). In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...

From these two axioms, it follows that for every g in G, the function which maps x in X to g·x is a bijective map from X to X. Therefore, one may alternatively define a group action of G on X as a group homomorphism from G into the symmetric group S_X. In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ...

In complete analogy, one can define a right group action of G on X as a function X × G → X by the two axioms:

1. x·(gh) = (x·g)·h
2. x·e = x

Note that the difference between left and right actions is only in the order in which a product like gh acts on x. For left actions h acts first followed by g, while for right actions g acts first followed by h. From a right action a left action can be constructed by composing with the inverse operation on the group. If r is a right action, then

is a left action, since

and

Therefore in the sequel, we consider only left group actions, since right actions add nothing.

Examples

• Every group G acts on G in two natural but essentially different ways: g·x = gx for all x in G, or g·x = gxg⁻¹ for all x in G.
• The symmetric group S_n and its subgroups act on the set { 1, ... , n } by permuting its elements: .
• The symmetry group of a polyhedron acts on the set of vertices of that 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 GL(n,R), special linear group SL(n,R), orthogonal group O(n,R), and special orthogonal group SO(n,R) are Lie groups which act on Rⁿ.
• The Galois group of a field extension E/F acts on the bigger field E. So does every subgroup of the Galois group.
• The additive group of the real numbers (R, +) acts on the phase space of "well-behaved" systems in classical mechanics (and in more general dynamical systems): if t is in R and x is in the phase space, then x describes a state of the system, and t·x 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 (R, +) acts on the set of real functions of a real variable with (g·f)(x) equal to e.g. f(x + g), f(x) + g, f(xe^g), f(x)e^g, f(x + g)e^g, or f(xe^g) + g, but not f(xe^g + g)
• The quaternions with modulus 1, as a multiplicative group, act on R³: for any such quaternion , the mapping f(x) = z x z^* is a counterclockwise rotation through an angle about an axis v; −z is the same rotation; see quaternions and spatial rotation.
• The isometries of the plane act on the set of 2D images and patterns, such as a wallpaper pattern. The definition can be made more precise by specifying what is meant by image or pattern, e.g. a function of position with values in a set of colors.
• More generally, a group of bijections g: V → V acts on the set of functions x: V → W by (gx)(v) = x(g⁻¹(v)) (or a restricted set of such functions that is closed under the group action). Thus a group of bijections of space induces a group action on "objects" in it.

In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... The symmetry group of an object (e. ... A polyhedron is a geometric shape which in mathematics is defined by three related meanings. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In geometry, a vertex (Latin: whirl, whirlpool; plural vertices) is a corner of a polygon (where two sides meet) or of a polyhedron (where three or more faces and an equal number of edges meet). ... In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. ... In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ... In mathematics, a Galois group is a group associated with a certain type of field extension. ... In mathematics, more specifically in abstract algebra, field extensions are the main object of study in field theory. ... In mathematics, the real numbers may be described informally in several different ways. ... Phase space of a dynamical system with focal stability. ... Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object -- a number, a function, a set, a space of one sort or another -- is well-behaved or not. ... Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Example of an Egyptian design with wallpaper group p4m A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. ...

Types of actions

The action of G on X is called

• transitive if for any two x, y in X there exists a g in G such that g·x = y;
  • sharply transitive if that g is unique;
• n-transitive if for any pairwise distinct x₁, ..., x_n and pairwise distinct y₁, ..., y_n there is a g in G such that g.x_k = y_k for 1 ≤ k ≤ n.
  • sharply n-transitive if there is exactly one such g.
• faithful (or effective) if for any two different g, h in G there exists an x in X such that g·x ≠ h·x; or equivalently, if for any g≠ e in G there exists an x in X such that g·x ≠ x.
• free if for any two different g, h in G and all x in X we have g·x ≠ h·x; or equivalently, if g·x = x for some x then g = e.
• regular (or simply transitive) if it is both transitive and free; this is equivalent to saying that for any two x, y in X there exists precisely one g in G such that g·x = y. In this case, X is known as a principal homogeneous space for G.

Every free action on a non-empty set is faithful. A group G acts faithfully on X if and only if the homomorphism G → Sym(X) has a trivial kernel. Thus, for a faithful action, G is isomorphic to a permutation group on X; specifically, G is isomorphic to its image in Sym(X). In mathematics, a principal homogeneous space, or G-torsor, for a group G is a set X on which G acts freely and transitively. ... In set theory, a set is called non-empty (or nonempty) if it contains at least one element, and is therefore not the empty set. ... It has been suggested that this article or section be merged with Logical biconditional. ... In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ... In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself); the relationship is often written as (G,M). ...

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) — a result known as Cayley's theorem. In group theory, Cayleys theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group on G. This can be understood as an example of the group action of G on the elements of G. A permutation of a...

If G does not act faithfully on X, one can easily modify the group to obtain a faithful action. If we define N = {g in G : g·x = x for all x in X}, then N is a normal subgroup of G; indeed, it is the kernel of the homomorphism G → Sym(X). The factor group G/N acts faithfully on X by setting (gN)·x = g·x. The original action of G on X is faithful if and only if N = {e}. In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ...

Orbits and stabilizers

Consider a group G acting on a set X. The orbit of a point x in X is the set of elements of X to which x can be moved by the elements of G. The orbit of x is denoted by Gx:

The defining properties of a group guarantee that the set of orbits of X under the action of G form a partition of X. The associated equivalence relation is defined by saying x ~ y if and only if there exists a g in G with g·x = y. 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, i.e. Gx = Gy. A partition of U into 6 blocks: a Venn diagram representation. ... In mathematics, an equivalence relation, denoted by an infix ~, is a binary relation on a set X that is reflexive, symmetric, and transitive. ... It has been suggested that this article or section be merged with Logical biconditional. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...

The set of all orbits of X under the action of G is written as X/G, and is called the quotient of the action; in geometric situations it may be called the orbit space.

If Y is a subset of X, we write GY for the set { g·y : y Y and g G}. We call the subset Y invariant under G if GY = Y (which is equivalent to GY ⊆ Y). In that case, G also operates on Y. The subset Y is called fixed under G if g·y = y for all g in G and all y in Y. Every subset that's fixed under G is also invariant under G, but not vice versa. A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ...

Every orbit is an invariant subset of X on which G acts transitively. The action of G on X is transitive if and only if all elements are equivalent, meaning that there is only one orbit.

For every x in X, we define the stabilizer subgroup of x (also called the isotropy group or little group) as 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 G → Sym(X) is given by the intersection of the stabilizers G_x for all x in X. In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ...

Orbits and stabilizers are not unrelated. For a fixed x in X, consider the map from G to X given by g g·x. The image of this map is the orbit of x and the coimage is the set of all left cosets of G_x. The standard quotient theorem of set theory then gives a natural bijection between G/G_x and Gx. Specifically, the bijection is given by hG_x h·x. This result is known as the orbit-stabilizer theorem. In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ... In mathematics, particularly in algebra, the coimage of a homomorphism f: A → B is the quotient coim f = A/ker f of the domain and kernel. ... In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G... A bijective function. ...

If G and X are finite then the orbit-stabilizer theorem, together with Lagrange's theorem, gives Lagranges theorem, in the mathematics of group theory, states that if G is a finite group and H is a subgroup of G, then the order (that is, the number of elements) of H divides the order of G. It is named after Joseph Lagrange. ...

This result is especially useful since it can be employed for counting arguments.

Note that if two elements x and y belong to the same orbit, then their stabilizer subgroups, G_x and G_y, are isomorphic. More precisely: if y = g·x, then G_y = gG_x g⁻¹. In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ...

A result closely related to the orbit-stabilizer theorem is Burnside's lemma: Burnsides lemma, sometimes also called Burnsides counting theorem, Pólyas formula, the Cauchy-Frobenius lemma or the Orbit-Counting Theorem, is a result in group theory which is often useful in taking account of symmetry when counting mathematical objects. ...

where X^g is 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.

Morphisms and isomorphisms between G-sets

If X and Y are two G-sets, we define a morphism from X to Y to be a function f : X → Y such that f(g.x) = g.f(x) for all g in G and all x in X. Morphisms of G-sets are also called equivariant maps or G-maps. In mathematics, an equivariant map is a function between two sets that commutes with the action of a group. ...

If such a function f is bijective, then its inverse is also a morphism, and we call f an isomorphism and the two G-sets X and Y are called isomorphic; for all practical purposes, they are indistinguishable in this case. In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...

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 G×S, where S is some set and G acts by left multiplication on the first coordinate.
• Every transitive G action is isomorphic to left multiplication by G on the set of left cosets of some subgroup H of G.

With this notion of morphism, the collection of all G-sets forms a category; this category is a topos. In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, a topos (plural topoi or toposes) is a type of category that behaves like the category of sheaves of sets on a topological space. ...

Continuous group actions

One often considers continuous group actions: the group G is a topological group, X is a topological space, and the map G × X → X is continuous with respect to the product topology of G × X. 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. In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ... In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ... For quotient spaces in linear algebra, see quotient space (linear algebra). ...

If G is a discrete group acting on a topological space X, the action is properly discontinuous if for any point x in X there is an open neighborhood U of x in X, such that the set of all for which consists of the identity only. 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 connected, 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 generalised in the book Topology and Groupoids referenced below to obtain the fundamental groupoid of the orbit space of a discontinuous action of 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 a symmetric square. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related branches of mathematics, an action of a group G on a topological space X is called properly discontinuous if every element of X has a neighborhood that moves outside itself under the action of any group element but the trivial element. ... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ...

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 GA = X. 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 G×X → X×X that sends is a proper map. In mathematics, a continuous function between topological spaces is called proper if inverse images of compact subsets are compact. ...

Strongly continuous group action and smooth vector

If is an action of a topological vector space V on an another topological vector space A, one says that it is strongly continuous if for all , the map is continuous with respect to the respective topologies.

Such an action induce an action on the space of continuous function on A by .

The space of smooth vector for the action α is the subspace of A of elements a such that is smooth, i.e. it is continuous and all derivatives are continuous.

Generalizations

One 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. In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ...

Instead of actions on sets, one 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. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ...

One can view a group G as a category with a single object in which every morphism is invertible. A group action is then nothing but a functor from G to the category of sets, and a group representation is a functor from G to the category of vector spaces. In analogy, an action of a groupoid is a functor from the groupoid to the category of sets or to some other category. In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ... In mathematics, the category K_Vect has all vector spaces over a fixed field K as objects and linear transformations as morphisms. ... In mathematics, especially in category theory and homotopy theory, a groupoid is a concept (first developed by Heinrich Brandt in 1926) that simultaneously generalises groups, equivalence relations on sets, and actions of groups on sets. ...

Without using the language of categories, one can extend the notion of a group action on a set X by studying as well its induced action on the power set of X. 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. See pattern groups. In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... In mathematics, the Mathieu groups are five finite simple groups discovered by the French mathematician Emile Léonard Mathieu. ... A finite geometry is any geometric system that has only a finite number of points. ...

See also

• Group with operators

In abstract algebra, a branch of pure mathematics, the algebraic structure group with operators or Ω-group is a group with a set of group endomorphisms. ...

References

• Brown, Ronald (2006). Topology and groupoids, Booksurge PLC, ISBN 1-4196-2722-8.
• Dummit, David, Richard Foote (2003). Abstract Algebra, (3rd ed.), Wiley. ISBN 0-471-43334-9.
• Rotman, Joseph (1995). An Introduction to the Theory of Groups, Graduate Texts in Mathematics 148, (4th ed.), Springer-Verlag. ISBN 0-387-94285-8.

• Weisstein, Eric W., Group Action at MathWorld.