# 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 AlgebraSET THEORY

Algebra "Algebraist" redirects here. For the novel by Iain M. Banks, see The Algebraist. The quadratic formula expresses the solution of the degree two equation in terms of its coefficients , where is not equal to Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values.[6] For example, in the letter is unknown, but the law of inverses can be used to discover its value: . , the letters and are variables, and the letter The word algebra is also used in certain specialized ways. A mathematician who does research in algebra is called an algebraist. How to distinguish between different meanings of "algebra" For historical reasons, the word "algebra" has several related meanings in mathematics, as a single word or with qualifiers. Algebra as a branch of mathematics can be any numbers whatsoever (except that cannot be Etymology History Early history of algebra History of 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

Set-builder notation Direct, ellipses, and informally specified sets A set is an unordered list of elements. The elements are also called set 'members'. Elements can be any mathematical entity. is a set holding the four numbers 3, 7, 15, and 31. is the set containing 'a','b', and 'c'. is the set of natural numbers. is the set of integers between 1 and 100 inclusive. all addresses on Pine Street is the set of all addresses on Pine Street. The ellipses means that the simplest interpretation should be applied for continuing a sequence. In the last example we use simple prose to describe what is in the set. The ellipses and simple prose approaches give the reader rules for building the set rather than directly presenting the elements. Formal set builder notation sets A set in set builder notation has three parts, a variable, a colon or vertical bar separator, and a logical predicate. or The x is taken to be a free variable. All values of x where the rule is true belong in the set. The is the set ,

Subgroup A proper subgroup of a group G is a subgroup H which is a proper subset of G (i.e. H ≠ G). The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. If H is a subgroup of G, then G is sometimes called an overgroup of H. The same definitions apply more generally when G is an arbitrary semigroup, but this article will only deal with subgroups of groups. The group G is sometimes denoted by the ordered pair (G, ∗), usually to emphasize the operation ∗ when G carries multiple algebraic or other structures. This article will write ab for a ∗ b, as is usual. Basic properties of subgroups G is the group , the integers mod 8 under addition. . Cosets and Lagrange's theorem If aH = Ha for every a in G, then H is said to be a normal subgroup. Example: Subgroups of Z8 Let G be the cyclic group Z8 whose elements are and whose group operation is addition modulo eight. Example: Subgroups of S4 (the symmetric group on 4 elements) 12 elements

Modular arithmetic Time-keeping on this clock uses arithmetic modulo 12. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. For a positive integer n, two integers a and b are said to be congruent modulo n, and written as History In the Third Century B.C.E., Euclid formalized, in his book Elements, the fundamentals of arithmetic, as well as showing his lemma, which he used to prove the Fundamental theorem of arithmetic. Congruence relation For example, because 38 − 14 = 24, which is a multiple of 12. The same rule holds for negative values: Equivalently, can also be thought of as asserting that the remainders of the division of both and by are the same. If then:

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.

Cardinality The cardinality of a set A is usually denoted | A |, with a vertical bar on each side; this is the same notation as absolute value and the meaning depends on context. Alternatively, the cardinality of a set A may be denoted by n(A), A, card(A), or # A. Comparing sets Definition 1: | A | = | B | For example, the set E = {0, 2, 4, 6, ...} of non-negative even numbers has the same cardinality as the set N = {0, 1, 2, 3, ...} of natural numbers, since the function f(n) = 2n is a bijection from N to E. Definition 2: | A | ≥ | B | A has cardinality greater than or equal to the cardinality of B if there exists an injective function from B into A. Definition 3: | A | > | B | A has cardinality strictly greater than the cardinality of B if there is an injective function, but no bijective function, from B to A. If | A | ≥ | B | and | B | ≥ | A | then | A | = | B | (Cantor–Bernstein–Schroeder theorem). Cardinal numbers Above, "cardinality" was defined functionally. For each .

Symmetrization In mathematics, symmetrization is a process that converts any function in n variables to a symmetric function in n variables. Conversely, anti-symmetrization converts any function in n variables into an antisymmetric function. 2 variables Let be a set and an Abelian group. is termed a symmetric map if for all The symmetrization of a map is the map Conversely, the anti-symmetrization or skew-symmetrization of a map The sum of the symmetrization and the anti-symmetrization is Thus, away from 2, meaning if 2 is invertible, such as for the real numbers, one can divide by 2 and express every function as a sum of a symmetric function and an anti-symmetric function. The symmetrization of a symmetric map is simply its double, while the symmetrization of an alternating map is zero; similarly, the anti-symmetrization of a symmetric map is zero, while the anti-symmetrization of an anti-symmetric map is its double. Bilinear forms a function is skew-symmetric if and only if it is symmetric (as

Newton's laws of motion First law: When viewed in an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by an external force.[2][3]Second law: F = ma. The vector sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration vector a of the object.Third law: When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body. The three laws of motion were first compiled by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), first published in 1687.[4] Newton used them to explain and investigate the motion of many physical objects and systems.[5] For example, in the third volume of the text, Newton showed that these laws of motion, combined with his law of universal gravitation, explained Kepler's laws of planetary motion. Overview Newton's first law Impulse

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