- Group theory
**Group theory**is a mathematical discipline, the part ofabstract algebra that studies thealgebraic structure s known as groups. The development of group theory sprang from three main sources:number theory , theory ofalgebraic equation s, andgeometry . The number-theoretic strand was started byLeonhard Euler and taken up by Gauss, who developedmodular arithmetic and considered additive and multiplicative groups related toquadratic field s. Early results aboutpermutation group s were obtained by Lagrange, Ruffini, and Abel in their quest for general solutions of polynomial equations of high degree. Galois coined the term “group” and established a connection between the nascent theory of groups and field theory, which is known asGalois theory . In geometry, groups first became important inprojective geometry and, later,non-Euclidean geometry .Felix Klein in hisErlangen program famously proclaimed group theory to be the organizing principle behind the very meaning of geometry.Groups manifest themselves as

symmetry group s of various physical systems, such ascrystal s and thehydrogen atom . Thus group theory and the closely relatedrepresentation theory have many applications inphysics andchemistry .The concept of a group is a central concept of abstract algebra: other algebraic structures, such as rings, fields, and

vector space s are elaborations of groups, which are endowed with additional operations. Groups recur throughout mathematics, and methods of group theory had a strong influence on ring theory and other parts of algebra.Linear algebraic group s andLie group s are two classes of groups whose theory has been tremendously advanced, and became the subject areas of their own.A central question of group theory throughout much of the last century was the

classification of finite simple groups . The result of a collaborative effort mostly from 1960-1980 and totaling more than ten thousand pages, it is one of the most important mathematical achievements of the 20th century.**History**There are three historical roots of group theory: the theory of

algebraic equation s,number theory andgeometry .Historically, the first use of groups to determine the solvability of

polynomial equation s was done byÉvariste Galois , in the 1830s. Investigations were pushed further, mainly in the guise ofpermutation group s, byArthur Cayley andAugustin Louis Cauchy . The second historical source for groups stems from geometrical situations. In an attempt to come to grips with possible geometries (such as euclidean, hyperbolic orprojective geometry ) using group theory,Felix Klein initiated theErlangen programme .Sophus Lie , in 1884, started using groups (now calledLie group s) attached to analytic problems. Thirdly, groups were (first implicitly and later explicitly) used inalgebraic number theory .The different scope of these early sources resulted in different notions of groups. The theory of groups was unified starting around 1880. Since, the impact of group theory has been ever growing, giving rise to the birth of

abstract algebra in the early 20th century,representation theory , and many more influential spin-off domains. Theclassification of finite simple groups is a vast body of work from the mid 20th century, classifying all the finitesimple group s.**Main classes of groups**The range of groups being considered has gradually expanded from finite

permutation group s and special examples ofmatrix group s to abstract groups that may be specified through a presentation by generators and relations.**Permutation groups**The first class of groups to undergo a systematic study was

permutation group s. Given any set "X" and a collection "G" of bijections of "X" into itself (known as "permutations") that is closed under compositions and inverses, "G" is a group acting on "X". If "X" consists of "n" elements and "G" consists of "all" permutations, "G" is thesymmetric group "S"_{"n"}; in general, "G" is asubgroup of the symmetric group of "X". An early construction due to Cayley exhibited any group as a permutation group, acting on itself ("X" = "G") by means of the leftregular representation .In many cases, the structure of a permutation group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for "n" ≥ 5, the

alternating group "A"_{"n"}is simple, i.e. does not admit any propernormal subgroup s. This fact plays a key role in the impossibility of solving a general algebraic equation of degree "n" ≥ 5 in radicals.**Matrix groups**The next important class of groups is given by "matrix groups", or

linear group s. Here "G" is a set consisting of invertible matrices of given order "n" over a field "K" that is closed under the products and inverses. Such a group acts on the "n"-dimensional vector space "K"^{"n"}bylinear transformation s. This action makes matrix groups conceptually similar to permutation groups, and geometry of the action may be usefully expoited to establish properties of the group "G".**Transformation groups**Permutation groups and matrix groups are special cases of

transformation group s: groups that act on a certain space "X" preserving its inherent structure. In the case of permutation groups, "X" is a set; for matrix groups, "X" is avector space . The concept of a transformation group is closely related with the concept of asymmetry group : transformation groups frequently consist of "all" transformations that preserve a certain structure. The theory of transformation groups forms a bridge connecting group theory withdifferential geometry . A long line of research, originating with Lie and Klein, considers group actions onmanifold s byhomeomorphism s ordiffeomorphism s. The groups themselves may be discrete or continuous.**Abstract groups**Most groups considered in the first stage of the development of group theory were "concrete", having been realized through numbers, permutations, or matrices. It was not until the late nineteenth century that the idea of an abstract group as a set with operations satisfying a certain system of axioms began to take hold. A typical way of specifying an abstract group is through a presentation by "generators and relations",

: $G\; =\; langle\; S|R\; angle.$

A significant source of abstract groups is given by the construction of a "factor group", or

quotient group , "G"/"H", of a group "G" by anormal subgroup "H".Class group s ofalgebraic number field s were among the earliest examples of factor groups, of much interest innumber theory . If a group "G" is a permutation group on a set "X", the factor group "G"/"H" is no longer acting on "X"; but the idea of an abstract group permits one not to worry about this discrepancy.The change of perspective from concrete to abstract groups makes it natural to consider properties of groups that are independent of a particular realization, or in modern language, invariant under

isomorphism , as well as the classes of group with a given such property:finite group s,periodic group s,simple group s,solvable group s, and so on. Rather than exploring properties of an individual group, one seeks to establish results that apply to a whole class of groups. The new paradigm was of paramount importance for the development of mathematics: it foreshadowed the creation ofabstract algebra in the works of Hilbert,Emil Artin ,Emmy Noether , and mathematicians of their school.**Topological and algebraic groups**An important elaboration of the concept of a group occurs if "G" is endowed with additional structure, notably, of a

topological space ,differentiable manifold , oralgebraic variety . If the group operations "m" (multiplication) and "i" (inversion),: $m:\; G\; imes\; G\; o\; G,\; (g,h)mapsto\; gh,\; quad\; i:G\; o\; G,\; gmapsto\; g^\{-1\},$

are compatible with this structure, i.e. are continuous, smooth or regular (in the sense of algebraic geometry) maps then "G" becomes a

topological group , aLie group , or analgebraic group . [*This process of imposing extra structure has been formalized through the notion of a*]group object in a suitable category. Thus Lie groups are group objects in the category of differentiable manifolds and affine algebraic groups are group objects in the category of affine algebraic varieties.The presence of extra structure relates these types of groups with other mathematical disciplines and means that more tools are available in their study. Topological groups form a natural domain for

abstract harmonic analysis , whereasLie group s (frequently realized as transformation groups) are the mainstays ofdifferential geometry and unitaryrepresentation theory . Certain classification questions that cannot be solved in general can be approached and resolved for special subclasses of groups. Thus, compact connected Lie groups have been completely classified. There is a fruitful relation between infinite abstract groups and topological groups: whenever a group "Γ" can be realized as a lattice in a topological group "G", the geometry and analysis pertaining to "G" yield important results about "Γ". A comparatively recent trend in the theory of finite groups exploits their connections with compact topological groups (profinite group s): for example, a single "p"-adic analytic group "G" has a family of quotients which are finite "p"-groups of various orders, and properties of "G" translate into the properties of its finite quotients.**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 nowrap|"g" • "h". 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"}, thefree group generated by "F" surjects onto the group "G". The kernel of this map is called subgroup of relations, generated by some subset "D". The presentation is usually denoted by nowrap begin〈"F" | "D" 〉nowrap end. For example, the group nowrap begin**Z**= 〈"a" | 〉nowrap end 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. [*harvnb|Schupp|Lyndon|2001*] It is particularly useful where finiteness 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 theirfundamental group s. 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. By relating the problem to

Turing machine s, one can show that there is in general noalgorithm solving this task. 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. [*harvnb|La Harpe|2000*] The first idea is made precise by means of theCayley graph , whose vertices correspond to group elements and edges correspond to right multiplication in the group. Given two elements, one constructs theword metric given by the length of the minimal path between the elements. A theorem of Milnor and Svarc then says that given a group "G" acting in a reasonable manner on ametric space "X", for example acompact manifold , then "G" is quasi-isometric (i.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. 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 other words, to every group element "g" is assigned anautomorphism "ρ"("g") such that nowrap begin"ρ"("g") ∘ "ρ"("h") = "ρ"("gh")nowrap end for any "h" in "G".This definition can be understood in two directions, both of which give rise to whole new domains of mathematics. [

*Such as*] 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. [group cohomology orequivariant K-theory .*In particular, if the representation is faithful.*] 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. These parts in turn are much more easily manageable than the whole "V" (viaSchur's lemma ).Given a group "G",

representation theory then asks what representations of "G" exist. There are several settings, and the employed methods and obtained results are rather different in every case:representation theory of finite groups and representations ofLie group s are two main subdomains of the theory. The totality of representations is governed by the group's characters. For example, Fourier polynomials can be interpreted as the characters of "U"(1), the group ofcomplex numbers ofabsolute value "1", acting on the "L"^{2}-space of periodic functions.**Connection of groups and symmetry**Given a structured object "X" of any sort, a

symmetry is a mapping of the object onto itself which preserves the structure. This occurs in many cases, for example

#If "X" is a set with no additional structure, a symmetry is a bijective map from the set to itself, giving rise topermutation group s.

#If the object "X" is a set of points in the plane with its metric structure or any othermetric space , a symmetry is a bijection of the set to itself which preserves the distance between each pair of points (anisometry ). The corresponding group is calledisometry group of "X".

#If insteadangle s are preserved, one speaks ofconformal map s. Conformal maps give rise toKleinian group s, for example.

#Symmetries are not restricted to geometrical objects, but include algebraic objects as well: the equation::"x"^{4}− 7"x"^{2}+ 12 = 0:has the solutions +2, −2, $+sqrt\{3\}$, and $-sqrt\{3\}$. Exchanging −2 and +2 and the two square roots determines a group, theGalois group belonging to the equation.The axioms of a group formalize the essential aspects of

symmetry . 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. 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. 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. Maps preserving the structure are then the

morphisms , and the symmetry group is theautomorphism group of the object in question.**Applications of group theory**Applications of group theory abound. Almost all structures in

abstract algebra are special cases of groups. Rings, for example, can be viewed asabelian group s (corresponding to addition) together with a second operation (corresponding to multiplication). 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). Thefundamental theorem of Galois theory provides a link betweenalgebraic field extension s and group theory. It gives an effective criterion for the solvability of polynomial equations in terms of the solvability of the correspondingGalois group . For example, "S"_{5}, thesymmetric group in 5 elements, is not solvable which implies that the generalquintic equation cannot be solved by radicals in the way equations of lower degree can. The theory, being one of the historical roots of group theory, is still fruitfully applied to yield new results in areas such asclass field theory .Algebraic topology is another domain which prominently associates groups to the objects the theory is interested in. There, groups are used to describe certain invariants oftopological space s. 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. For example, thefundamental group "counts" how many paths in the space are essentially different. ThePoincaré conjecture , proved in 2002/2003 byPerelman is a prominent application of this idea. The influence is not unidirectional, though. For example, algebraic topology makes use ofEilenberg-MacLane space s which are spaces with prescribedhomotopy groups . Similarlyalgebraic K-theory stakes in a crucial way onclassifying space s of groups. Finally, the name of thetorsion subgroup of an infinite group shows the legacy of topology in group theory.Algebraic geometry andcryptography likewise uses group theory in many ways.Abelian varieties have been introduced above. 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. [*For example the*] The one-dimensional case, namelyHodge conjecture (in certain cases).elliptic curve s is studied in particular detail. They are both theoretically and practically intriguing. [*See the*] Very large groups of prime order constructed in Elliptic-Curve Cryptography serve forBirch-Swinnerton-Dyer conjecture , one of themillennium problem spublic key cryptography . 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 thediscrete logarithm very hard to calculate. One of the earliest encryption protocols, Caesar's cipher, may also be interpreted of a (very easy) group operation. In another direction, toric varieties are algebraic varieties acted on by atorus . Toroidal embeddings have recently led to advances inalgebraic geometry , in particularresolution of singularities . [*Citation | last1=Abramovich | first1=Dan | last2=Karu | first2=Kalle | last3=Matsuki | first3=Kenji | last4=Wlodarczyk | first4=Jaroslaw | title=Torification and factorization of birational maps | id=MathSciNet | id = 1896232 | year=2002 | journal=*]Journal of the American Mathematical Society | issn=0894-0347 | volume=15 | issue=3 | pages=531–572Algebraic number theory is a special case of group theory, thereby following the rules of the latter. For example, Euler's product formula:$egin\{align\}sum\_\{ngeq\; 1\}frac\{1\}\{n^s\}\; =\; prod\_\{p\; ext\{\; prime\; frac\{1\}\{1-p^\{-s\; \backslash end\{align\}!$captures the fact that any integer decomposes in a unique way into primes. The failure of this statement for more general rings gives rise toclass group s andregular prime s, which feature in Kummer's treatment ofFermat's Last Theorem .*The concept of the

Lie group (named after mathematicianSophus Lie ) is important in the study ofdifferential equations andmanifold s; they describe the symmetries of continuous geometric and analytical structures. Analysis on these and other groups is calledharmonic analysis .Haar measure s, that is integrals invariant under the translation in a Lie group, are used forpattern recognition and otherimage processing techniques. [*Citation | last1=Lenz | first1=Reiner | title=Group theoretical methods in image processing | publisher=*]Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Computer Science | isbn=978-0-387-52290-6 | year=1990 | volume=413*In

combinatorics , the notion ofpermutation group and the concept of group action are often used to simplify the counting of a set of objects; see in particularBurnside's lemma .*The presence of the 12-

periodicity in thecircle of fifths yields applications ofelementary group theory inmusical set theory .*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 group s, since these representations often point the way to the "possible" physical theories. 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 assigned point groups can then be used to determine physical properties (such as polarity and chirality), spectroscopic properties (particularly useful forRaman spectroscopy andInfrared spectroscopy ), and to construct molecular orbitals.**See also***

Group (mathematics)

*Glossary of group theory

*List of group theory topics **Notes****References*** | year=1991 | volume=126

* | year=1969 | journal=Communications of the Association for Computing Machinery | issn=0001-0782 | volume=12 | pages=3–12

*Connell, Edwin, " [*http://www.math.miami.edu/~ec/book/ Elements of Abstract and Linear Algebra.*] " Free online textbook.

*

* | year=1986 | journal=Mathematics Magazine | issn=0025-570X | volume=59 | issue=4 | pages=195–215

*

*cite book | author=Livio, M. | title= The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry | publisher=Simon & Schuster | year=2005 | id=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.

*

*cite book | author=Rotman, Joseph | title=An introduction to the theory of groups | location=New York | publisher=Springer-Verlag | year=1994 | id=ISBN 0-387-94285-8 A standard contemporary reference.

* | year=1994

*

*cite book | author=Scott, W. R. | title= Group Theory | location=New York | publisher=Dover | year=1987 | origyear=1964 | id=ISBN 0-486-65377-3 Inexpensive and fairly readable, but somewhat dated in emphasis, style, and notation.

* | year=1972

* |journal= Bull. Amer. Math. Soc. (N.S.) | issn =0273-0979 |volume=43 | year= 2006 | pages=305--364 This shows an advantage of the generalisation from group togroupoid .**External links*** [

*http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Abstract_groups.html History of the abstract group concept*]

* [*http://www.bangor.ac.uk/r.brown/hdaweb2.htm 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.

*Wikimedia Foundation.
2010.*