# Group (mathematics)

Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; Point groups are used to help understand symmetry phenomena in molecular chemistry; and Poincaré groups can express the physical symmetry underlying special relativity. One of the most familiar groups is the set of integers Z which consists of the numbers ..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ...,[3] together with addition. The following properties of integer addition serve as a model for the abstract group axioms given in the definition below. The integers, together with the operation +, form a mathematical object belonging to a broad class sharing similar structural aspects. Closure Associativity Related:  Abstract Algebra

Dynkin diagram The term "Dynkin diagram" can be ambiguous. In some cases, Dynkin diagrams are assumed to be directed, in which case they correspond to root systems and semi-simple Lie algebras, while in other cases they are assumed to be undirected, in which case they correspond to Weyl groups; the and directed diagrams yield the same undirected diagram, correspondingly named In this article, "Dynkin diagram" means directed Dynkin diagram, and undirected Dynkin diagrams will be explicitly so named. Classification of semisimple Lie algebras The fundamental interest in Dynkin diagrams is that they classify semisimple Lie algebras over algebraically closed fields. Dropping the direction on the graph edges corresponds to replacing a root system by the finite reflection group it generates, the so-called Weyl group, and thus undirected Dynkin diagrams classify Weyl groups. Related classifications The root lattice generated by the root system, as in the E8 lattice. corresponds to Example: A2 for or

History of group theory Early 19th century The earliest study of groups as such probably goes back to the work of Lagrange in the late 18th century. However, this work was somewhat isolated, and 1846 publications of Cauchy and Galois are more commonly referred to as the beginning of group theory. The theory did not develop in a vacuum, and so 3 important threads in its pre-history are developed here. Development of permutation groups One foundational root of group theory was the quest of solutions of polynomial equations of degree higher than 4. An early source occurs in the problem of forming an equation of degree m having as its roots m of the roots of a given equation of degree n > m. A common foundation for the theory of equations on the basis of the group of permutations was found by mathematician Lagrange (1770, 1771), and on this was built the theory of substitutions. Ruffini (1799) attempted a proof of the impossibility of solving the quintic and higher equations. Felix Klein Sophus Lie

Orbifold notation In geometry, orbifold notation (or orbifold signature) is a system, invented by William Thurston and popularized by the mathematician John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advantage of the notation is that it describes these groups in a way which indicates many of the groups' properties: in particular, it describes the orbifold obtained by taking the quotient of Euclidean space by the group under consideration. Groups representable in this notation include the point groups on the sphere ( ), the frieze groups and wallpaper groups of the Euclidean plane ( ), and their analogues on the hyperbolic plane ( Definition of the notation The following types of Euclidean transformation can occur in a group described by orbifold notation: All translations which occur are assumed to form a discrete subgroup of the group symmetries being described. Each symbol corresponds to a distinct transformation: Good orbifolds John H.

Orthogonality The line segments AB and CD are orthogonal to each other. The concept of orthogonality has been broadly generalized in mathematics, science, and engineering, especially since the beginning of the 16th century. Much of the generalizing has taken place in the areas of mathematical functions, calculus and linear algebra. Etymology The word comes from the Greek ὀρθός (orthos), meaning "upright", and γωνία (gonia), meaning "angle". Mathematics Definitions A set of vectors is called pairwise orthogonal if each pairing of them is orthogonal. In certain cases, the word normal is used to mean orthogonal, particularly in the geometric sense as in the normal to a surface. A vector space with a bilinear form generalizes the case of an inner product. Euclidean vector spaces Note however that there is no correspondence with regards to perpendicular planes, because vectors in subspaces start from the origin (by the definition of a vector subspace). Orthogonal functions where

Homeomorphism group Properties and Examples There is a natural group action of the homeomorphism group of a space on that space. If this action is transitive, then the space is said to be homogeneous. Topology As with other sets of maps between topological spaces, the homeomorphism group can be given a topology, such as the compact-open topology (in the case of regular, locally compact spaces), making it into a topological group. In the category of topological spaces with homeomorphisms, object groups are exactly homeomorphism groups. Mapping class group In geometric topology especially, one considers the quotient group obtained by quotienting out by isotopy, called the mapping class group: The MCG can also be interpreted as the 0th homotopy group, . In some applications, particularly surfaces, the homeomorphism group is studied via this short exact sequence, and by first studying the mapping class group and group of isotopically trivial homeomorphisms, and then (at times) the extension.

Field theory (mathematics) Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields. Intuitively, a field is a set F that is a commutative group with respect to two compatible operations, addition and multiplication, with "compatible" being formalized by distributivity, and the caveat that the additive identity (0) has no multiplicative inverse (one cannot divide by 0). Closure of F under addition and multiplication For all a, b in F, both a + b and a · b are in F (or more formally, + and · are binary operations on F). Associativity of addition and multiplication For all a, b, and c in F, the following equalities hold: a + (b + c) = (a + b) + c and a · (b · c) = (a · b) · c. Commutativity of addition and multiplication For all a and b in F, the following equalities hold: a + b = b + a and a · b = b · a. Existence of additive and multiplicative identity elements Distributivity of multiplication over addition a + bζ

Group isomorphism Definition and notation Given two groups (, ∗) and (, ), a group isomorphism from (, ∗) to (, ) is a bijective group homomorphism from to . Spelled out, this means that a group isomorphism is a bijective function such that for all and in it holds that The two groups (, ∗) and (, ) are isomorphic if there exists an isomorphism from one to the other. Often shorter and simpler notations can be used. Sometimes one can even simply write = . Conversely, given a group (, ∗), a set , and a bijection , we can make a group (, ) by defining If = and = ∗ then the bijection is an automorphism (q.v.). Intuitively, group theorists view two isomorphic groups as follows: For every element g of a group G, there exists an element h of H such that h 'behaves in the same way' as g (operates with other elements of the group in the same way as g). An isomorphism of groups may equivalently be defined as an invertible morphism in the category of groups, where invertible here means has a two-sided inverse. . Define .

Isomorphism The group of fifth roots of unity under multiplication is isomorphic to the group of rotations of the pentagon under composition. In mathematics, an isomorphism, from the Greek: ἴσος isos "equal", and μορφή morphe "shape", is a homomorphism (or more generally a morphism) that admits an inverse.[note 1] Two mathematical objects are isomorphic if an isomorphism exists between them. An automorphism is an isomorphism whose source and target coincide. The interest of isomorphisms lies in the fact that two isomorphic objects cannot be distinguished by using only the properties used to define morphisms; thus isomorphic objects may be considered the same as long as one considers only these properties and their consequences. In topology, where the morphisms are continuous functions, isomorphisms are also called homeomorphisms or bicontinuous functions. Isomorphisms are formalized using category theory. Examples Logarithm and exponential Let be the additive group of real numbers. for all

Heap (mathematics) which satisfies the para-associative law the identity law A group can be regarded as a heap under the operation . makes H into a group with identity e and inverse . Whereas the automorphisms of a single object form a group, the set of isomorphisms between two isomorphic objects naturally forms a heap, with the operation (here juxtaposition denotes composition of functions). If then the following structure is a heap: As noted above, any group becomes a heap under the operation One important special case: are integers, we can set to produce a heap. to be the identity of a new group on the set of integers, with the operation and inverse A pseudoheap or pseudogroud satisfies the partial para-associative condition[2] A semiheap or semigroud is required to satisfy only the para-associative law but need not obey the identity law.[3] An example of a semigroud that is not in general a groud is given by M a ring of matrices of fixed size with and for all a and b. is reflexive (idempotence) and anti-symmetric.

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