In mathematics, a symmetry group describes all symmetries of objects. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or 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 Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property 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). In Mathematics, a permutation group is a group G whose elements are Permutations of a given set M, and whose group operation In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that 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. In several fields of Mathematics the term permutation is used with different but closely related meanings In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of In Mathematics, in Matrix theory, a permutation matrix is a square (01-matrix that has exactly one entry 1 in each row and each column and 0's elsewhere Basis vector redirects here For basis vector in the context of crystals see Crystal structure.

Given an equilateral triangle, the counterclockwise rotation by 120° around the center of the triangle "acts" on the set of vertices of the triangle by mapping every vertex to another one. Properties The area of an equilateral triangle with sides of length a\\! A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation

Definition

If G is a group and X is a set, then a (left) group action of G on X is a binary function

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. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a binary function, or function of two variables, is a function which takes two inputs 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 The group G is said to act on X (on the left).

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. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property 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, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying

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

The difference between left and right actions is in the order in which a product like gh acts on x. For a left action h acts first and is followed by g, while for a right action g acts first and is 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

Similarly, any left action can be converted into a right action. Therefore in the sequel we consider only left group actions, since right actions add nothing new.

Examples

• The trivial action for any group G is defined by g·x=x for all g in G and all x in X; that is, the whole group G induces the identity permutation on X. In Mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that
• 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. An exponential notation is commonly used for the right-action variant of the latter case: x^g = g⁻¹xg. The latter action is often called the conjugation action. In Abstract algebra, an inner automorphism of a group G is a function f: G &rarr G
• The symmetric group S_n and its subgroups act on the set { 1, . In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of . .  , n } by permuting its elements
• The symmetry group of a polyhedron acts on the set of vertices of that polyhedron. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is What is a polyhedron? We can at least say that a polyhedron is built up from different kinds of element or entity each associated with a different number of dimensions
• 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. 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 (called vectors) that informally speaking may be scaled and added In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real . . ) 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. In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation In Mathematics, the special linear group of degree n over a field F is the set of n × n matrices with In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group
• The Galois group of a field extension E/F acts on the bigger field E. 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. 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. In Mathematics, the real numbers may be described informally in several different ways In Mathematics and Physics, a phase space, introduced by Willard Gibbs in 1901 is a Space in which all possible states of a System Mathematicians (and those in related sciences very frequently speak of whether a mathematical object &mdash a Number, a function, a set, a space Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects Dynamical systems theory is an area of Applied mathematics used to describe the behavior of complex Dynamical systems usually by employing Differential
• 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. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. Unit quaternions provide a convenient mathematical notation for representing Orientations and Rotations of objects in three dimensions
• The isometries of the plane act on the set of 2D images and patterns, such as a wallpaper pattern. A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern based on the 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.

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; it is equivalent to regularity defined below.
• 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. A 2-transitive action is also called doubly transitive, a 3-transitive action is also called triply transitive, and so on.
  • sharply n-transitive if there is exactly one such g.
• faithful (or effective) if for any two distinct 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 or semiregular if for any two distinct g, h in G and all x in X we have g·x ≠ h·x; or equivalently, if g·x = x for some x implies 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 or as a G-torsor. In Mathematics, a principal homogeneous space, or torsor, for a group G is a set X on which G acts freely and
• locally free if G is a topological group, and there is a neighbourood U of e in G such that the restriction of the action to U is free; that is, if g·x = x for some x and some g in U then g = e. In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the

Every free action on a non-empty set is faithful. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members A group G acts faithfully on X if and only if the homomorphism G → Sym(X) has a trivial kernel. ↔ 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 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 permutation group is a group G whose elements are Permutations of a given set M, and whose group operation

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, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a Subgroup

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). In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. The factor group G/N acts faithfully on X by setting (gN)·x = g·x. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G The original action of G on X is faithful if and only if N = {e}.

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. In Mathematics, a partition of a set X is a division of X into non-overlapping " parts " or " blocks " The associated equivalence relation is defined by saying x ~ y if and only if there exists a g in G with g·x = y. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" ↔ 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. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X e. Gx = Gy.

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.

Every orbit is an invariant subset of X on which G acts transitively. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. 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. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of 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 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

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. In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage In Mathematics, particularly in algebra, the coimage of a Homomorphism f:  A  → B In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH The standard quotient theorem of set theory then gives a natural bijection between G/G_x and Gx. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property Specifically, the bijection is given by hG_x h·x. This result is known as the orbit-stabilizer theorem.

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

This result is especially useful since it can be employed for counting arguments. Lagrange's theorem, in the Mathematics of Group theory, states that for any Finite group G, the order (number of elements of

Note that if two elements x and y belong to the same orbit, then their stabilizer subgroups, G_x and G_y, are isomorphic (or conjugate). 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 In Mathematics, especially Group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class More precisely: if y = g·x, then G_y = gG_x g⁻¹. Points with conjugate stabilizer subgroups are said to have the same orbit-type.

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

where X^g is the set of points fixed by g. Burnside's lemma, sometimes also called Burnside's counting theorem, the Cauchy-Frobenius lemma or the orbit-counting theorem, is a result in Group 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.

The set of formal differences of finite G-sets forms a ring called the Burnside ring, where addition corresponds to disjoint union, and multiplication to Cartesian product. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, the Burnside ring of a Finite group is an algebraic construction that encodes the different ways the group can act on finite sets In Set theory, a disjoint union (or discriminated union) is a modified union operation which indexes the elements according to which set they originated Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory.

Group actions and groupoids

The notion of group action can be put in a broader context by using the associated `action groupoid' associated to the group action, thus allowing techniques from groupoid theory such as presentations and fibrations. In Mathematics, especially in Category theory and Homotopy theory Further the stabilisers 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 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, 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.

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, or a bijective function is a function f from a set X to a set Y with the property In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective

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. In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of

With this notion of morphism, the collection of all G-sets forms a category; this category is a topos. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, a topos (plural "topoi" or "toposes" is a type of category that behaves like the category of sheaves of sets

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. In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural 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. In Topology, a discrete space is a particularly simple example of a Topological space or similar structure one in which the points are " isolated " 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. In Topology and related areas of Mathematics, a quotient space (also called an identification space) is intuitively speaking the result of identifying The above statements about isomorphisms for regular, free and transitive actions are no longer valid for continuous group actions.

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. Topological spaces are mathematical structures that allow the formal definition of 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 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. In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism 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. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of 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. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. This allows calculations such as the fundamental group of a symmetric square.

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. In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation This does not define bijective maps and equivalence relations however.

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. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets 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, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of

One can view a group G as a category with a single object in which every morphism is invertible. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and 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 Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose Morphisms are In Mathematics, especially Category theory, the category K-Vect has all Vector spaces over a fixed field K as objects 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, especially in Category theory and Homotopy theory

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. In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) 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. In the Mathematical field of Group theory, the Mathieu groups, named after the French mathematician Émile Léonard Mathieu, are five finite simple A finite geometry is any geometric system that has only a finite number of points.

See also

• Group with operators
• Act, the action of a monoid on a set. In Abstract algebra, a branch of pure Mathematics, the Algebraic structure group with operators or Ω- group is a group with a In Theoretical computer science, a semiautomaton is is an automaton having only an input and no output In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation
• Gain graph

References

• Aschbacher, Michael (2000), Finite Group Theory, Cambridge University Press, MR1777008, ISBN 978-0-521-78675-1 
• Brown, Ronald (2006). A gain graph is a graph whose edges are labelled invertibly, or orientably, by elements of a group G. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many Topology and groupoids, Booksurge PLC, ISBN 1-4196-2722-8.

• Categories and groupoids, P.J. Higgins, downloadable reprint of van Nostrand Notes in Mathematics, 1971, which deal with applications of groupoids in group theory and topology.
• 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.  

• Eric W. Weisstein, Group Action at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W