Groups

Group theory

This box: view • talk • edit

In mathematics, group theory is the field that studies the algebraic structures known as groups. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, 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

Groups were first introduced in mathematics in the 19th century by mathematicians like Galois, Abel and Cauchy in their quest for general solutions of polynomial equations. The formal abstract definition—as described in the group article—was, however, not introduced until the 20th century. 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

Groups have become a central object in the study of abstract algebra, and are building blocks of more elaborate algebraic structures such as rings, fields, and vector spaces, and recur throughout mathematics. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Group theory has many applications in physics and chemistry, and is potentially applicable in any situation characterized by symmetry. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Chemistry (from Egyptian kēme (chem meaning "earth") is the Science concerned with the composition structure and properties 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

The classification of finite simple groups is a major mathematical achievement of the 20th century. The classification of the finite simple groups, also called the enormous theorem is believed to classify all finite simple groups.

Contents 1 History 2 Groups 2.1 Basic definitions 2.2 Finiteness conditions 2.3 Abelian groups 2.4 Normal series 2.5 Groups with additional structure 3 Combinatorial and geometric group theory 4 Representation of groups 5 Connection of groups and symmetry 6 Applications of group theory 7 Miscellany 8 Notes 9 References 10 See also 11 External links

History

There are three historical roots of group theory: the theory of algebraic equations, number theory and geometry. In Mathematics, an algebraic equation over a given field is an Equation of the form P = Q where P and Q Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Euler, Gauss, Lagrange, Abel and French mathematician Galois were early researchers in the field of group theory. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German Niels Henrik Abel (August 5 1802 &ndash April 6 1829 was a noted Norwegian Mathematician who proved the impossibility of solving the Quintic equation Galois is honored as the first mathematician linking group theory and field theory, with the theory that is now called Galois theory. In Mathematics, more specifically in Abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory ^[1]

An early source occurs in the problem of forming an mth-degree equation having as its roots m of the roots of a given nth-degree equation (m < n). For simple cases the problem goes back to Hudde (1659). Johannes (van Waveren Hudde ( April 23, 1628, Amsterdam - April 15, 1704, Amsterdam was a Burgomaster (mayor of Amsterdam Saunderson (1740) noted that the determination of the quadratic factors of a biquadratic expression necessarily leads to a sextic equation, and Le Sœur (1748) and Waring (1762 to 1782) still further elaborated the idea. Nicholas Saunderson (1682&ndash19 April 1739 was an English Scientist and Mathematician. Edward Waring (1736 – August 15, 1798) was an English Mathematician who was born in Old Heath (near Shrewsbury) ^[1]

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. In several fields of Mathematics the term permutation is used with different but closely related meanings He discovered that the roots of all resolvents (résolvantes, réduites) which he examined are rational functions of the roots of the respective equations. To study the properties of these functions he invented a Calcul des Combinaisons. The contemporary work of Vandermonde (1770) also foreshadowed the coming theory. Alexandre-Théophile Vandermonde ( 28 February 1735 – 1 January 1796) was a French Musician and Chemist who ^[1]

Ruffini (1799) attempted a proof of the impossibility of solving the quintic and higher equations. Paolo Ruffini ( September 22, 1765 – May 9, 1822) was an Italian Mathematician and Philosopher. In Mathematics, a quintic equation is a Polynomial Equation of degree five Ruffini distinguished what are now called intransitive and transitive, and imprimitive and primitive groups, and (1801) uses the group of an equation under the name l'assieme delle permutazioni. In Mathematics, the word transitive admits at least three distinct meanings A group G acts transitively on a In Mathematics, a Permutation group G acting on a set X is called primitive if G preserves no nontrivial partition of X He also published a letter from Abbati to himself, in which the group idea is prominent. Pietro Abbati Marescotti ( 1 September 1768 – 7 May 1842) was a Italian Mathematician. ^[1]

Galois found that if are the n roots of an equation, there is always a group of permutations of the r's such that (1) every function of the roots invariable by the substitutions of the group is rationally known, and (2), conversely, every rationally determinable function of the roots is invariant under the substitutions of the group. Galois also contributed to the theory of modular equations and to that of elliptic functions. In Mathematics, a modular equation is an Algebraic equation satisfied by moduli, in the sense of Moduli problem. In Complex analysis, an elliptic function is a function defined on the Complex plane which is periodic in two directions (a Doubly-periodic His first publication on group theory was made at the age of eighteen (1829), but his contributions attracted little attention until the publication of his collected papers in 1846 (Liouville, Vol. XI). ^[1]

Arthur Cayley and Augustin Louis Cauchy were among the first to appreciate the importance of the theory, and to the latter especially are due a number of important theorems. Arthur Cayley ( August 16 1821 - January 26 1895) was a British Mathematician. The subject was popularized by Serret, who devoted section IV of his algebra to the theory; by Camille Jordan, whose Traité des Substitutions is a classic; and to Eugen Netto (1882), whose Theory of Substitutions and its Applications to Algebra was translated into English by Cole (1892). Joseph Alfred Serret ( August 30[[ 819]] - March 2, 1885) was a French Mathematician who was born in Paris France Marie Ennemond Camille Jordan ( January 5 1838 &ndash January 22 1922) was a French Mathematician, known both for his foundational Eugen Otto Erwin Netto (30 June 1848 - 13 May 1919 was a German mathematician Other group theorists of the nineteenth century were Bertrand, Charles Hermite, Frobenius, Leopold Kronecker, and Emile Mathieu. Joseph Louis François Bertrand ( March 11, 1822 – April 5, 1900, born and died in Paris was a French Mathematician who Charles Hermite (ʃaʁl ɛʁˈmit ( December 24, 1822 &ndash January 14, 1901) was a French Mathematician who did Ferdinand Georg Frobenius ( October 26, 1849 – August 3, 1917) was a German Mathematician, best-known for his contributions Leopold Kronecker ( December 7, 1823 – December 29, 1891) was a German Mathematician and Logician who argued Émile Léonard Mathieu ( May 15, 1835, Metz – October 19, 1890, Nancy) was a French Mathematician ^[1]

Walther von Dyck was the first (in 1882) to define a group in the full abstract sense of this entry. Walther Franz Anton von Dyck ( December 6, 1856 - November 5, 1934) was a German Mathematician.

The study of what are now called Lie groups, and their discrete subgroups, as transformation groups, started systematically in 1884 with Sophus Lie; followed by work of Killing, Study, Schur, Maurer, and Cartan. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics, a discrete group is a group G equipped with the Discrete topology. 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 Marius Sophus Lie (liː as "Lee" ( 17 December 1842 - 18 February 1899) was a Norwegian -born Mathematician. Wilhelm Karl Joseph Killing ( May 10 1847 &ndash February 11 1923) was a German Mathematician who made important contributions Eduard Study ( March 23, 1862 &ndash January 6, 1930) was a German Mathematician known for work on Invariant theory Schur is a surname and may refer to Alexander Schur (born 1971 German footballer Issai Schur (1875-1941 Lithuanian-German-Israeli mathematician For the composer see Ludwig Wilhelm Maurer. Ludwig Maurer (1859 – 1927 was a German Mathematician. Élie Joseph Cartan ( 9 April 1869 &ndash 6 May 1951) was an influential French Mathematician, who did fundamental The discontinuous (discrete group) theory was built up by Felix Klein, Lie, Poincaré, and Charles Émile Picard, in connection in particular with modular forms and monodromy. In Mathematics, a discrete group is a group G equipped with the Discrete topology. Felix Christian Klein ( 25 April 1849 &ndash 22 June 1925) was a German Mathematician, known for his work in Group Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician Charles Émile Picard (usually referred to simply as Émile Picard) ( July 24, 1856 - December 12, 1941) was a leading French In Mathematics, a modular form is a (complex Analytic function on the Upper half-plane satisfying a certain kind of Functional equation and In Mathematics, monodromy is the study of how objects from Mathematical analysis, Algebraic topology and algebraic and Differential geometry

The classification of finite simple groups is a vast body of work from the mid 20th century, classifying all the finite simple groups. The classification of the finite simple groups, also called the enormous theorem is believed to classify all finite simple groups. In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. SIMPLE Group Limited is a conglomeration of separately run companies that each specialised in a particular area of Tax Planning

Other important contributors to group theory include Emil Artin, Emmy Noether, Sylow, and many others. Emil Artin ( March 3, 1898, in Vienna – December 20, 1962, in Hamburg) was an Austrian Mathematician Amalie Emmy Noether, ˈnøːtɐ (23 March 1882 – 14 April 1935 was a German Mathematician known for her groundbreaking contributions to Abstract algebra and Peter Ludwig Mejdell Sylow ( 12 December 1832 &ndash 7 September 1918) was a Norwegian Mathematician, who proved foundational

Alfred Tarski proved elementary group theory undecidable. Alfred Tarski ( January 14, 1901, Warsaw, Russian ruled Poland – October 26, 1983, Berkeley California In Computability theory and Computational complexity theory, a decision problem is a question in some Formal system with a yes-or-no answer depending on ^[2]

University of Florida Graduate Research Professor John Griggs Thompson and European mathematician Jacques Tits won the 2008 Abel Prize for their contributions to group theory. The Abel Prize is an international prize presented annually by the King of Norway to one or more outstanding Mathematicians The prize is named after Norwegian

Groups

Basic definitions

Main article: Group (mathematics)

A group (G, •) is a set G closed under a binary operation • satisfying the following 3 axioms:

• Associativity: For all a, b and c in G, (a • b) • c = a • (b • c). 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 set is said to be closed under some operation if the operation on members of the set produces a member of the set In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject In Mathematics, associativity is a property that a Binary operation can have
• Identity element: There exists an e∈G such that for all a in G, e • a = a • e = a. 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
• Inverse element: For each a in G, there is an element b in G such that a • b = b • a = e, where e is an identity element. In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to

Basic examples for groups are the integers Z with addition operation, or rational numbers without zero Q\{0} with multiplication. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions More generally, for any ring R, the units in R form a multiplicative group. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i In Mathematics and Group theory the term multiplicative group refers to one of the following concepts depending on the context any group \scriptstyle\mathfrak See the group article for an illustration of this definition and for further examples. 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 Groups include, however, much more general structures than the above. Group theory is concerned with proving abstract statements about groups, regardless of the actual nature of element and the operation of the groups in question.

A subset H ⊂ G is a subgroup if the restriction of • to H is a group operation on H. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of It is called normal, if left and right cosets agree, i. In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH e. gH = Hg for all g in G. Normal subgroups play a distinguished role by virtue of the fact that the collection of cosets of a normal subgroup N in a group G naturally inherits a group structure, enabling the formation of the quotient group, usually denoted G/N (also called a factor group). In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G The Butterfly lemma is a technical result on the lattice of subgroups of a group. In Mathematics, the butterfly lemma or Zassenhaus lemma, named after Hans Julius Zassenhaus, is a technical result on the Lattice of subgroups In Mathematics, the lattice of subgroups of a group G is the lattice whose elements are the Subgroups of G with the

A group homomorphism is a map f : G → H between two groups that preserves the structure imposed by the operation, i. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function e.

f(a•b) = f(a) • f(b).

Bijective (in-, surjective) maps are isomorphisms of groups (mono-, epimorphisms, respectively). In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism. In Category theory an epimorphism (also called an epic morphism or an epi) is a Morphism f: X &rarr Y which The kernel ker(f) is always a normal subgroup of the group. 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 For f as above, the fundamental theorem on homomorphisms relates the structure of G and H, and of the kernel and image of the homomorphism, namely

G / ker(f) ≅ im(f)

Groups together with group homomorphisms form a category. In Abstract algebra, the Fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, relates the structure of two objects 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, a category is a fundamental and abstract way to describe mathematical entities and their relationships

In universal algebra, groups are generally treated as algebraic structures of the form (G, •, e, ⁻¹), i. Universal algebra (sometimes called general algebra) is the field of Mathematics that studies Algebraic structures themselves not examples ("models" e. the identity element e and the map that takes every element a of the group to its inverse a⁻¹ are treated as integral parts of the formal definition of a group.

Finiteness conditions

The order |G| (or o(G)) of a group is the cardinality of G. In Group theory, a branch of Mathematics, the term order is used in two closely related senses the order of a group is In Mathematics, the cardinality of a set is a measure of the "number of elements of the set" If the order |G| is (in-)finite, then G itself is called (in-)finite. In Mathematics, a finite group is a group which has finitely many elements An important class is the group of permutations or symmetric groups of N letters, denoted S_N. In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying Cayley's theorem exhibits any finite group G as a subgroup of the symmetric group on G. In Group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a Subgroup In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying The theory of finite groups is very rich. Lagrange's theorem states that the order of any subgroup H of a finite group G divides the order of G. Lagrange's theorem, in the Mathematics of Group theory, states that for any Finite group G, the order (number of elements of A partial converse is given by the Sylow theorems: if pⁿ is the greatest power of a prime p dividing the order of a finite group G, then there exists a subgroup of order pⁿ, and the number of these subgroups is also known. In Mathematics, specifically Group theory, the Sylow theorems, named after Ludwig Sylow, form a partial converse to Lagrange's theorem, which A projective limit of finite groups is called profinite^[3]. In Mathematics, the inverse limit (also called the projective limit) is a construction which allows one to "glue together" several related objects the precise In Mathematics, profinite groups are Topological groups that are in a certain sense assembled from Finite groups they share many properties with their finite An important profinite group, fundamental for p-adic analysis, class field theory, and l-adic cohomology is the ring of p-adic integers and the profinite completion of Z, respectively

and ^[4]

Most of the facts from finite groups can be generalized directly to the profinite case. In Mathematics, p -adic analysis is a branch of Number theory that deals with the Mathematical analysis of functions of P-adic numbers In Mathematics, class field theory is a major branch of Algebraic number theory. In Mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897 In Mathematics, profinite groups are Topological groups that are in a certain sense assembled from Finite groups they share many properties with their finite ^[5]

Certain conditions on chains of subgroups, parallel to the notion of Noetherian and Artinian rings, allow to deduce further properties. The ascending chain condition (ACC and descending chain condition (DCC are finiteness properties satisfied by certain algebraic structures most importantly ideals In Abstract algebra, a Noetherian ring is a ring that satisfies the Ascending chain condition on ideals. In Abstract algebra, an Artinian ring is a ring that satisfies the Descending chain condition on ideals. For example the Krull-Schmidt theorem states that a group satisfying certain finiteness conditions for chains of its subgroups, can be uniquely written as a finite direct product of indecomposable subgroups. In Mathematics, the Krull-Schmidt theorem states that a group G subjected to certain finiteness conditions of chains of Subgroups

Another, yet slightly weaker, level of finiteness is the following: a subset A of G is said to generate the group if any element h can be written as the product of elements of A. In Abstract algebra, a generating set of a group G is a Subset S such that every element of G can be expressed as the A group is said to be finitely generated if it is possible to find a finite subset A generating the group. In Abstract algebra, a generating set of a group G is a Subset S such that every element of G can be expressed as the Finitely generated groups are in many respects as well-treatable as finite groups.

Abelian groups

The category of groups can be subdivided in several ways. In Mathematics, the category Grp has the class of all groups for objects and Group homomorphisms for Morphisms As such A particularly well-understood class of groups are the so-called abelian (in honor of Niels Abel, or commutative) groups, i. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the Niels Henrik Abel (August 5 1802 &ndash April 6 1829 was a noted Norwegian Mathematician who proved the impossibility of solving the Quintic equation In Mathematics, commutativity is the ability to change the order of something without changing the end result e. the ones satisfying

a • b = b • a for all a, b in G.

Another way of saying this is that the commutator

[a, b] := a⁻¹b⁻¹ab

equals the identity element. In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative. A non-abelian group is a group that is not abelian. Even more particular, cyclic groups are the groups generated by a single element. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In Abstract algebra, a generating set of a group G is a Subset S such that every element of G can be expressed as the Being either isomorphic to Z or to Z_n, the integers modulo n, they are always abelian. The word modulo (Latin with respect to a modulus of ___ is the Latin Ablative of Modulus which itself means "a small measure Any finitely generated abelian group is known to be a direct sum of groups of these two types. In Abstract algebra, an Abelian group ( G,+ is called finitely generated if there exist finitely many elements x 1 In Abstract algebra, an Abelian group ( G,+ is called finitely generated if there exist finitely many elements x 1 The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction The category of abelian groups is an abelian category. In Mathematics, an abelian category is a category in which Morphisms and objects can be added and in which kernels and Cokernels exist In fact, abelian groups serve as the prototype of abelian categories. A converse is given by Mitchell's embedding theorem. Mitchell's embedding theorem, also known as the Freyd-Mitchell theorem is a mathematical result about abelian categories; it states that these categories while rather abstractly

Normal series

Most of the notions developed in group theory are designed to tackle non-abelian groups. There are several notions designed to measure how far a group is from being abelian. The commutator subgroup (or derived group) is the subgroup generated by commutators [a, b], whereas the center is the subgroup of elements that commute with every other group element. In Mathematics, more specifically in Abstract algebra, the commutator subgroup or derived subgroup of a group is the Subgroup In Abstract algebra, the center of a group G is the set Z ( G) of all elements in G which commute with all the

Given a group G and a normal subgroup N of G, denoted N ⊲ G, there is an exact sequence:

1 → N → G → H → 1,

where 1 denotes the trivial group and H is the quotient G/N. In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group In Mathematics, a trivial group is a group consisting of a single element This permits the decomposition of G into two smaller pieces. The other way round, given two groups N and H, a group G fitting into an exact sequence as above is called an extension of H by N. Given H and N there are many different group extensions G, which leads to the extension problem. There is always at least one extension, called the trivial extension, namely the direct sum G = N ⊕ H, but usually there are more. The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction For example, the Klein four-group is a non-trivial extension of Z₂ by Z₂. In Mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V) is the group Z2 This is a first glimpse of homological algebra and Ext functors. Homological algebra is the branch of Mathematics which studies homology in a general algebraic setting In Mathematics, the Ext functors of Homological algebra are Derived functors of Hom functors They were first used in Algebraic topology ^[6]

Many properties for groups, for example being a finite group or a p-group are stable under extensions and sub- and quotient groups, i. In Mathematics, a finite group is a group which has finitely many elements In Mathematics, given a Prime number p, a p -group is a Periodic group in which each element has a power of p e. if N and H have the property, then so does G and vice versa. This kind of information is therefore preserved while breaking it into pieces by means of exact sequences. If this process has come to an end, i. e. if a group G does not have any (non-trivial) normal subgroups, G is called simple. SIMPLE Group Limited is a conglomeration of separately run companies that each specialised in a particular area of Tax Planning

The name simple can be slightly misleading insofar that the group does not have to be easy to understand. The following concepts remedy this problem: repeatedly taking normal subgroups (if they exist) leads to normal series:

1 = G₀ ⊲ G₁ ⊲ . Mathematical induction is a method of Mathematical proof typically used to establish that a given statement is true of all Natural numbers It is done by proving that In Mathematics, a subgroup series is a chain of Subgroups 1 = A_0 \leq A_1 \leq \cdots \leq A_n = G . . ⊲ G_n = G,

i. e. any G_i is a normal subgroup of the next one G_i+1. A group is solvable (or soluble) if it has a normal series all of whose quotients are abelian. In the history of Mathematics, the origins of Group theory lie in the search for a proof of the general unsolvability of Quintic and higher equations finally Imposing further commutativity constraints on the quotients G_i+1 / G_i, one obtains central series which lead to nilpotent groups. In Mathematics, especially in the fields of Group theory and Lie theory, a central series is a kind of Normal series of Subgroups or In Group theory, a nilpotent group is a group having a special property that makes it "almost" abelian, through repeated application of the They are an approximation of abelian groups in the sense that

[. . . [[g₁, g₂], g₃] . . . , g_n]=1

for all choices of group elements g_i.

There may be distinct normal series for a group G. If it is impossible to refine a given series by inserting further normal subgroups, it is called composition series. In Abstract algebra, a composition series provides a way to break up an algebraic structure such as a group or a module, into simple pieces By the Jordan-Hölder theorem any two composition series of a given group are equivalent. In Abstract algebra, a composition series provides a way to break up an algebraic structure such as a group or a module, into simple pieces ^[7]

Groups with additional structure

Many groups come with an additional structure. This idea is made formal by the notion of a group object, i. In Mathematics, group objects are certain generalizations of groups which are built on more complicated structures than sets. e. an object X in a category C together with a morphism

m : X × X → X

satisfying all the axioms of an abstract group. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and This way, abstract group theory and the structure embodied by the category are intertwined.

Probably the most prominent case are Lie groups, i. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group e. manifolds M together with a smooth multiplication map as above, as well as an inverse map i : M → M and a distinguished point 1 ∈ M, such that m(x, i(x)) = 1. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability The starting example is the general linear group GL_n, over R or C. In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation Being an open subset of the (affine) space of all matrices, it is a manifold, and the multiplication maps are smooth, for they are given by polynomials in the entries of the matrix. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations

The general paradigm, that a group structure on any kind of object greatly simplifies the description of the object, peaks in the classification of all compact Lie groups, a task that would be impossible for compact manifolds instead. In Mathematics, a compact ( topological, often understood group is a Topological group whose Topology is Compact. (Locally) compact Lie groups play a particularly important rôle. In Mathematics, a locally compact group is a Topological group G which is locally compact as a Topological space.

The parallel situation in algebraic geometry deals with algebraic groups, i. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In Algebraic geometry, an algebraic group (or group variety) is a group that is an Algebraic variety, such that the multiplication and inverse e. algebraic varieties possessing a compatible (i. This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety e. given by regular maps) group structure. In Complex analysis, see Holomorphic function. In Mathematics, a regular function in the sense of Algebraic geometry They fall into two regimes: projective and affine ones. This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety The group underlying the former can be shown to be necessarily abelian, therefore they are called abelian varieties^[8], which is in marked contrast to the affine case. In Mathematics, particularly in Algebraic geometry, Complex analysis and Number theory, an Abelian variety is a projective algebraic variety Affine algebraic groups are necessarily groups of matrices, for example the SL_n, which are commutative only in trivial cases. In Mathematics, a matrix group is a group G consisting of invertible matrices over some field K, usually fixed In Mathematics, the special linear group of degree n over a field F is the set of n × n matrices with ^[9]

A third important class are topological groups. In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the Lie group and algebraic groups are topological groups, by forgetting the differential or variety structure. Profinite group are examples of compact groups. In Mathematics, a compact ( topological, often understood group is a Topological group whose Topology is Compact.

Combinatorial and geometric group theory

Groups can be described in different ways. Finite groups can be described by writing down the group table consisting of all possible multiplications g • h. A Cayley table, after the 19th century British Mathematician Arthur Cayley, describes the structure of a A more important way of defining a group is by generators and relations, also called the presentation of a group. Given any set F of generators {g_i}_{i ∈ I}, the free group generated by F surjects onto the group G. In Mathematics, a group G is called free if there is a Subset S of G such that any element of G can be The kernel of this map is called subgroup of relations, generated by some subset D. The presentation is usually denoted by 〈F | D 〉. For example, the group Z = 〈a | 〉 can be generated by one element a (equal to +1 or −1) and no relations, because n·1 never equals 0 unless n is zero. A string consisting of generator symbols is called a word.

Combinatorial group theory studies groups from the perspective of generators and relations. Geometric group theory is an area in Mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and ^[10] It is particularly useful where finitness assumptions are satisfied, for example finitely generated groups, or finitely presented groups (i. e. in addition the relations are finite). The area makes use of the connection of graphs via their fundamental groups. In Mathematics and Computer science, a graph is the basic object of study in Graph theory. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. For example, one can show that every subgroup of a free group is free.

There are several natural questions arising from giving a group by its presentation. The word problem asks whether two words are effectively the same group element. In Mathematics, especially in the area of Abstract algebra known as Combinatorial group theory, the word problem for a recursively presented By relating the problem to Turing machines, one can show that there is in general no algorithm solving this task. Turing machines are basic abstract symbol-manipulating devices which despite their simplicity can be adapted to simulate the logic of any Computer Algorithm In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation An equally difficult problem is, whether two groups given by different presentations are actually isomorphic. For example Z can also be presented by

〈x, y | xyxyx = 1〉

and it is not obvious (but true) that this presentation is isomorphic to the standard one above.

Geometric group theory attacks these problems from a geometric viewpoint, either by viewing groups as geometric objects, or by finding suitable geometric objects a group acts on. Geometric group theory is an area in Mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and ^[11] The first idea is made precise by means of the Cayley graph, whose vertices correspond to group elements and edges correspond to right multiplication in the group. In Mathematics, the Cayley graph, also known as the Cayley colour graph, is the graph that encodes the structure of a Discrete group. Given two elements, one constructs the word metric given by the length of the minimal path between the elements. In Group theory, a word metric on a group G is a way to measure distance between any two elements of G. A theorem of Milnor and Svarc then says that given a group G acting in a reasonable manner on a metric space X, for example a compact manifold, then G is quasi-isometric (i. John Willard Milnor (b February 20, 1931 in Orange New Jersey) is an American Mathematician known for his work in Differential In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined See also Classification of manifolds#Point-set In Mathematics, a closed manifold is type of Topological space, namely a compact This is a glossary of some terms used in Riemannian geometry and Metric geometry &mdash it doesn't cover the terminology of Differential topology. e. looks similar from the far) to the space X.

Representation of groups

Saying that a group G acts on a set X means that every element defines a bijective map on a set in a way compatible with the group structure. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. When X has more structure, it is useful to restrict this notion further: a representation of G on a vector space V is a group homomorphism

ρ : G → GL(V),

where GL(V) consists of the invertible linear transformations of V. 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, the general linear group of degree n is the set of n × n invertible matrices, together with the operation In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In other words, to every group element g is assigned an automorphism ρ(g) such that ρ(g) ∘ ρ(h) = ρ(gh) for any h in G. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself

This definition can be understood in two directions, both of which give rise to whole new domains of mathematics. ^[12] On the one hand, it may yield new information about the group G: often, the group operation in G is abstractly given, but via ρ, it corresponds to the multiplication of matrices, which is very explicit. In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix ^[13] On the other hand, given a well-understood group acting on a complicated object, this simplifies the study of the object in question. For example, if G is finite, it is known that V above decomposes into irreducible parts. Maschke's theorem is a theorem in Group representation theory which concerns the decomposition of representations of a Finite group into irreducible pieces In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of These parts in turn are much more easily manageable than the whole V (via Schur's lemma). In Mathematics, Schur's lemma is an elementary but extremely useful statement in Representation theory of groups and Algebras In the group case

Given a group G, representation theory then asks what representations of G exist. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of There are several settings, and the employed methods and obtained results are rather different in every case: representation theory of finite groups and representations of Lie groups are two main subdomains of the theory. In Mathematics, Representation theory is a technique for analyzing abstract groups in terms of groups of Linear transformations See the article on 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 totality of representations is governed by the group's characters. This article refers to the use of the term character theory in mathematics for the media studies definition see Character theory (Media. For example, Fourier polynomials can be interpreted as the characters of U(1), the group of complex numbers of absolute value 1, acting on the L²-space of periodic functions. In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions In Mathematics, the unitary group of degree n, denoted U( n) is the group of n × n unitary matrices Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. In Mathematics, the Lp and ℓp spaces are spaces of p-power integrable functions, and corresponding

Connection of groups and symmetry

Main article: Symmetry group

Given a structured object X of any sort, a symmetry is a mapping of the object onto itself which preserves the structure. 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 occurs in many cases, for example

1. If X is a set with no additional structure, a symmetry is a bijective map from the set to itself, giving rise to permutation groups. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Mathematics, a permutation group is a group G whose elements are Permutations of a given set M, and whose group operation
2. If the object X is a set of points in the plane with its metric structure or any other metric space, a symmetry is a bijection of the set to itself which preserves the distance between each pair of points (an isometry). In Mathematics, a metric or distance function is a function which defines a Distance between elements of a set. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined For the Mechanical engineering and Architecture usage see Isometric projection. The corresponding group is called isometry group of X. In Mathematics, the isometry group of a Metric space is the set of all isometries from the metric space onto itself with the Function composition
3. If instead angles are preserved, one speaks of conformal maps. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called In Mathematics, a conformal map is a function which preserves Angles In the most common case the function is between domains in the Complex plane Conformal maps give rise to Kleinian groups, for example. In Mathematics, a Kleinian group, named after Felix Klein, is a finitely generated Discrete group &Gamma of orientation preserving conformal
4. Symmetries are not restricted to geometrical objects, but include algebraic objects as well: the equation

x⁴ − 7x² + 12 = 0

has the solutions +2, −2, , and . Exchanging −2 and +2 and the two square roots determines a group, the Galois group belonging to the equation. In Mathematics, a Galois group is a group associated with a certain type of Field extension.

The axioms of a group formalize the essential aspects of symmetry. 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 Symmetries form a group: they are closed because if you take a symmetry of an object, and then apply another symmetry, the result will still be a symmetry. In Mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set The identity keeping the object fixed is always a symmetry of an object. Existence of inverses is guaranteed by the undoing the symmetry and the associativity comes from the fact that symmetries are functions on a space, and composition of functions are associative.

Frucht's theorem says that every group is the symmetry group of some graph. In Mathematics and Computer science, a graph is the basic object of study in Graph theory. So every abstract group is actually the symmetries of some explicit object.

The saying of "preserving the structure" of an object can be made precise by working in a category. In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships Maps preserving the structure are then the morphisms, and the symmetry group is the automorphism group of the object in question. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself

Applications of group theory

Applications of group theory abound. Almost all structures in abstract algebra are special cases of groups. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules Rings, for example, can be viewed as abelian groups (corresponding to addition) together with a second operation (corresponding to multiplication). In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real Therefore group theoretic arguments underlie large parts of the theory of those entities.

Galois theory uses groups to describe the symmetries of the roots of a polynomial (or more precisely the automorphisms of the algebras generated by these roots). In Mathematics, more specifically in Abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory The fundamental theorem of Galois theory provides a link between algebraic field extensions and group theory. In Mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of Field extensions In its most basic In Abstract algebra, a Field extension L / K is called algebraic if every element of L is algebraic over K, i It gives an effective criterion for the solvability of polynomial equations in terms of the solvability of the corresponding Galois group. In Mathematics, a Galois group is a group associated with a certain type of Field extension. For example, S₅, the symmetric group in 5 elements, is not solvable which implies that the general quintic equation cannot be solved by radicals in the way equations of lower degree can. In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying In Mathematics, a quintic equation is a Polynomial Equation of degree five The theory, being one of the historical roots of group theory, is still fruitfully applied to yield new results in areas such as class field theory. In Mathematics, class field theory is a major branch of Algebraic number theory.

Algebraic topology is another domain which prominently associates groups to the objects the theory is interested in. Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories There, groups are used to describe certain invariants of topological spaces. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. They are called "invariants" because they are defined in such a way that they do not change if the space is subjected to some deformation. Topological equivalence redirects here see also Topological equivalence (dynamical systems. For example, the fundamental group "counts" how many paths in the space are essentially different. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. The Poincaré conjecture, proved in 2002/2003 by Perelman is a prominent application of this idea. In Mathematics, the Poincaré conjecture (French pwɛ̃kaʀe is a Theorem about the characterization of the three-dimensional sphere among Perelman is a Surname and may refer to any of the following people Bob Perelman, poet Chaim Perelman, philosopher The influence is not unidirectional, though. For example, algebraic topology makes use of Eilenberg-MacLane spaces which are spaces with prescribed homotopy groups. In Mathematics, an Eilenberg-MacLane space is a special kind of Topological space that can be regarded as a building block for Homotopy theory. In Mathematics, homotopy groups are used in Algebraic topology to classify Topological spaces The base point preserving maps from an n -dimensional Similarly algebraic K-theory stakes in a crucial way on classifying spaces of groups. In Mathematics, algebraic K-theory is an advanced part of Homological algebra concerned with defining and applying a sequence K n In Mathematics, a classifying space BG in Homotopy theory of a Topological group G is the quotient of a Weakly contractible Finally, the name of the torsion subgroup of an infinite group shows the legacy of topology in group theory. In the theory of Abelian groups the torsion subgroup AT of an abelian group A is the Subgroup of A consisting of all elements

Algebraic geometry and cryptography likewise uses group theory in many ways. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with Cryptography (or cryptology; from Greek grc κρυπτός kryptos, "hidden secret" and grc γράφω gráphō, "I write" Abelian varieties have been introduced above. In Mathematics, particularly in Algebraic geometry, Complex analysis and Number theory, an Abelian variety is a projective algebraic variety The presence of the group operation yields additional information which makes these varieties particularly accessible. They also often serve as a test for new conjectures. ^[14] The one-dimensional case, namely elliptic curves is studied in particular detail. In Mathematics, an elliptic curve is a smooth, projective Algebraic curve of genus one on which there is a specified point O They are both theoretically and practically intriguing. ^[15] Very large groups of prime order constructed in Elliptic-Curve Cryptography serve for public key cryptography. Elliptic curve cryptography (ECC is an approach to Public-key cryptography based on the algebraic structure of Elliptic curves over Finite fields The use Public-key cryptography, also known as asymmetric cryptography, is a form of Cryptography in which the key used to encrypt a message differs from the key Cryptographical methods of this kind benefit from the flexibility of the geometric objects, hence their group structures, together with the complicated structure of these groups, which make the discrete logarithm very hard to calculate. In Mathematics, specifically in Abstract algebra and its applications discrete logarithms are group-theoretic analogues of ordinary Logarithms One of the earliest encryption protocols, Caesar's cipher, may also be interpreted of a (very easy) group operation. In Cryptography, a Caesar cipher, also known as a Caesar's cipher, the shift cipher, Caesar's code or Caesar shift, is one of the In another direction, toric varieties are algebraic varieties acted on by a torus. In Mathematics and Theoretical physics, toric geometry is a set of methods in Algebraic geometry in which certain Complex manifolds are visualized This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar Toroidal embeddings have recently led to advances in algebraic geometry, in particular resolution of singularities. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In Algebraic geometry, the problem of resolution of singularities asks whether any Algebraic variety has a non-singular model (a non-singular variety ^[16]

Algebraic number theory would not exist without group theory. In Mathematics, algebraic number theory is a major branch of Number theory which studies the Algebraic structures related to Algebraic integers For example, Euler's product formula

captures the fact that any integer decomposes in a unique way into primes. In Number theory, an Euler product is an Infinite product expansion indexed by Prime numbers p, of a Dirichlet series. In Number theory, the fundamental theorem of arithmetic (or unique-prime-factorization theorem) states that every Natural number greater than 1 can be written In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 The failure of this statement for more general rings gives rise to class groups and regular primes, which feature in Kummer's treatment of Fermat's last theorem. In Abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an Integral domain in which every nonzero Proper In Mathematics, the extent to which Unique factorization fails in the ring of integers of an Algebraic number field (or more generally any Dedekind domain In Number theory, a regular prime is a certain kind of Prime number. Ernst Eduard Kummer ( 29 January 1810 - 14 May 1893) was a German Mathematician. Fermat's Last Theorem is the name of the statement in Number theory that It is impossible to separate any power higher than the second into two like

• The concept of the Lie group (named after mathematician Sophus Lie) is important in the study of differential equations and manifolds; they describe the symmetries of continuous geometric and analytical structures. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group Marius Sophus Lie (liː as "Lee" ( 17 December 1842 - 18 February 1899) was a Norwegian -born Mathematician. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be Analysis on these and other groups is called harmonic analysis. Harmonic analysis is the branch of Mathematics that studies the representation of functions or signals as the superposition of basic Waves It investigates and generalizes Haar measures, that is integrals invariant under the translation in a Lie group, are used for pattern recognition and other image processing techniques. In Mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of Locally compact topological groups and subsequently define Pattern recognition is a sub-topic of Machine learning. It is "the act of taking in raw data and taking an action based on the category of the data" Image processing is any form of Signal processing for which the input is an image such as photographs or frames of video the output of image processing can be either an image ^[17]

• In combinatorics, the notion of permutation group and the concept of group action are often used to simplify the counting of a set of objects; see in particular Burnside's lemma. Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects In several fields of Mathematics the term permutation is used with different but closely related meanings Burnside's lemma, sometimes also called Burnside's counting theorem, the Cauchy-Frobenius lemma or the orbit-counting theorem, is a result in Group

• The presence of the 12-periodicity in the circle of fifths yields applications of elementary group theory in musical set theory. Periodicity is the quality of occurring at regular intervals or periods (in Time or Space) and can occur in different contexts A Clock marks In Music theory, the circle of fifths (or '''circle of fourths''') shows the relationships among the twelve tones of the Chromatic scale, their corresponding In Mathematics, a group G,*> is defined as a set G and a Binary operation on G, called product and denoted Musical set theory provides concepts for categorizing musical objects and describing their relationships

• An understanding of group theory is also important in physics and chemistry and material science. In physics, groups are important because they describe the symmetries which the laws of physics seem to obey. Physicists are very interested in group representations, especially of Lie groups, since these representations often point the way to the "possible" physical theories. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group Examples of the use of groups in physics include: Standard Model, Gauge theory, Lorentz group, Poincaré group

• In chemistry, groups are used to classify crystal structures, regular polyhedra, and the symmetries of molecules. The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations In Physics (and mathematics the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting In Physics and Mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime Chemistry (from Egyptian kēme (chem meaning "earth") is the Science concerned with the composition structure and properties Molecular symmetry in Chemistry describes the Symmetry present in Molecules and the classification of molecules according to their symmetry The assigned point groups can then be used to determine physical properties (such as polarity and chirality), spectroscopic properties (particularly useful for Raman spectroscopy and Infrared spectroscopy), and to construct molecular orbitals. In Physics, polarity is a description of an attribute typically a binary attribute (one with two values or a Vector (a direction The term chiral (pronounced /ˈkaɪɹ(əl̩/ is used to describe an object that is non- superimposable on its mirror image Raman spectroscopy (pronounced S— is a spectroscopic technique used in Condensed matter physics and Chemistry to study vibrational rotational and Infrared spectroscopy (IR spectroscopy is the subset of Spectroscopy that deals with the Infrared region of the Electromagnetic spectrum.

Miscellany

In philosophy, Ernst Cassirer related group theory to the theory of perception of Gestalt Psychology. Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language Ernst Cassirer ( July 28, 1874 &ndash April 13, 1945) was a German Jewish Philosopher. Gestalt psychology (also Gestalt of the Berlin School) is a theory of mind and brain that proposes that the operational principle of the brain is holistic He took the Perceptual Constancy of that psychology as analogous to the invariants of group theory.

Notes

1. ^ ^{a b c d e f} Smith, D. E. , History of Modern Mathematics, Project Gutenberg, 1906. Project Gutenberg, abbreviated as PG, is a volunteer effort to Digitize, archive and distribute Cultural works
2. ^ Tarski, Alfred (1953) "Undecidability of the elementary theory of groups" in Tarski, Mostowski, and Raphael Robinson Undecidable Theories. Raphael Mitchel Robinson ( November 2 1911, National City California - January 27 1995. North-Holland: 77-87.
3. ^ Shatz 1972
4. ^ These two groups play a central role for maximal abelian extension of number fields, see Kronecker-Weber theorem
5. ^ For example the Sylow theorems. In Abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. In Mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) Field extension of the In Algebraic number theory, the Kronecker–Weber theorem states that every finite Abelian extension of the field of Rational numbers Q, or in
6. ^ Weibel 1994
7. ^ This can be shown using the Schreier refinement theorem. In Mathematics, the Schreier refinement theorem of Group theory states that any two Normal series of Subgroups ending with the Trivial group
8. ^ Mumford 1970
9. ^ Borel 1991
10. ^ Schupp & Lyndon 2001
11. ^ La Harpe 2000
12. ^ Such as group cohomology or equivariant K-theory. In Abstract algebra, Homological algebra, Algebraic topology and Algebraic number theory, as well as in applications to Group theory proper
13. ^ In particular, if the representation is faithful. In Mathematics, especially in the area of Abstract algebra known as Representation theory, a faithful representation ρ of a group G
14. ^ For example the Hodge conjecture (in certain cases). The Hodge conjecture is a major unsolved problem in Algebraic geometry which relates the Algebraic topology of a Non-singular complex Algebraic
15. ^ See the Birch-Swinnerton-Dyer conjecture, one of the millenium problems
16. ^ Abramovich, Dan; Karu, Kalle; Matsuki, Kenji & Wlodarczyk, Jaroslaw (2002), “Torification and factorization of birational maps”, Journal of the American Mathematical Society 15 (3): 531–572, MR1896232, ISSN 0894-0347 
17. ^ Lenz, Reiner (1990), Group theoretical methods in image processing, vol. In Mathematics, the Birch and Swinnerton-Dyer conjecture relates the rank of the Abelian group of points over a Number field of an Elliptic The Clay Mathematics Institute (CMI is a private Non-profit foundation, based in Cambridge, Massachusetts. The Journal of the American Mathematical Society, often referred to by its acronym JAMS, is a mathematics journal published quarterly by the American Mathematical Society Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many An International Standard Serial Number ( ISSN) is a unique eight-digit number used to identify a print or electronic Periodical publication. 413, Lecture Notes in Computer Science, Berlin, New York: Springer-Verlag, ISBN 978-0-387-52290-6 

References

• Borel, Armand (1991), Linear algebraic groups, vol. Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic Armand Borel ( 21 May 1923 &ndash 11 August 2003) was a Swiss Mathematician, born in La Chaux-de-Fonds, and was 126 (2nd ed. ), Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, MR1102012, ISBN 978-0-387-97370-8 
• Cannon, John J. Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many (1969), “Computers in group theory: A survey”, Communications of the Association for Computing Machinery 12: 3–12, MR0290613, ISSN 0001-0782 
• Connell, Edwin, Elements of Abstract and Linear Algebra. Free online textbook. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many An International Standard Serial Number ( ISSN) is a unique eight-digit number used to identify a print or electronic Periodical publication.
• Frucht, R. (1939), “Herstellung von Graphen mit vorgegebener abstrakter Gruppe”, Compositio Mathematica 6: 239–50, ISSN 0010-437X, <http://www.numdam.org/numdam-bin/fitem?id=CM_1939__6__239_0> 
• Kleiner, Israel (1986), “The evolution of group theory: a brief survey”, Mathematics Magazine 59 (4): 195–215, MR863090, ISSN 0025-570X 
• La Harpe, Pierre de (2000), Topics in geometric group theory, University of Chicago Press, ISBN 978-0-226-31721-2 
• Livio, M. An International Standard Serial Number ( ISSN) is a unique eight-digit number used to identify a print or electronic Periodical publication. Mathematics Magazine is a refereed bimonthly publication of the Mathematical Association of America. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many An International Standard Serial Number ( ISSN) is a unique eight-digit number used to identify a print or electronic Periodical publication. The University of Chicago Press is the largest University press in the United States (2005). The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry. Simon & Schuster. ISBN 0-7432-5820-7.   A pleasant read, explaining the importance of group theory and how its symmetries point to symmetries in physics and other sciences. Conveys well the practical value of group theory.
• Mumford, David (1970), Abelian varieties, Oxford University Press, ISBN 978-0-19-560528-0, OCLC 138290 
• Rotman, Joseph (1994). David Bryant Mumford (born 11 June 1937) is a Mathematician known for distinguished work in Algebraic geometry, and then for research into The OCLC Online Computer Library Center is according to its website a "nonprofit membership computer library service and research organization dedicated to the public purpose An introduction to the theory of groups. New York: Springer-Verlag. ISBN 0-387-94285-8.   A standard contemporary reference.
• Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge University Press, MR1269324, ISBN 978-0-521-55987-4 
• Schupp, Paul E. Cambridge University Press (known colloquially as CUP is a Publisher given a Royal Charter by Henry VIII in 1534 Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many & Lyndon, Roger C. (2001), Combinatorial group theory, Berlin, New York: Springer-Verlag, ISBN 978-3-540-41158-1 
• Scott, W. Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic R. [1964] (1987). Group Theory. New York: Dover. ISBN 0-486-65377-3.   Inexpensive and fairly readable, but somewhat dated in emphasis, style, and notation.
• Shatz, Stephen S. (1972), Profinite groups, arithmetic, and geometry, Princeton University Press, MR0347778, ISBN 978-0-691-08017-8 

See also

• group (mathematics)
• Glossary of group theory
• List of group theory topics

External links

• History of the abstract group concept
• Higher dimensional group theory This presents a view of group theory as level one of a theory which extends in all dimensions, and has applications in homotopy theory and to higher dimensional nonabelian methods for local-to-global problems. The Princeton University Press is an independent publisher with close connections to Princeton University. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many 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 A group ( G, • is a set G closed under a Binary operation • satisfying the following 3 Axioms: This is a list of group theory topics, by Wikipedia page to provide an overview of the topic and to allow those interested to use the "Related changes" feature to

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic