TheInfoList

OR:

In
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as
rings Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
,
fields Fields may refer to: Music * Fields (band), an indie rock band formed in 2006 * Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song b ...
, and
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macro ...
s and the hydrogen atom, and three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
have many important applications in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, chemistry, and materials science. Group theory is also central to
public key cryptography Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic al ...
. The early
history of group theory The history of group theory, a mathematical domain studying groups in their various forms, has evolved in various parallel threads. There are three historical roots of group theory: the theory of algebraic equations, number theory and geometry. Jose ...
dates from the 19th century. One of the most important mathematical achievements of the 20th century was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 2004, that culminated in a complete
classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else i ...
.

# History

Group theory has three main historical sources:
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
, the theory of
algebraic equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
s, and
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
. The number-theoretic strand was begun by
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
, and developed by Gauss's work on
modular arithmetic In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his boo ...
and additive and multiplicative groups related to
quadratic field In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 a ...
s. Early results about permutation groups were obtained by Lagrange, Ruffini, and
Abel Abel ''Hábel''; ar, هابيل, Hābīl is a Biblical figure in the Book of Genesis within Abrahamic religions. He was the younger brother of Cain, and the younger son of Adam and Eve, the first couple in Biblical history. He was a shepherd ...
in their quest for general solutions of polynomial equations of high degree.
Évariste Galois Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, ...
coined the term "group" and established a connection, now known as
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
, between the nascent theory of groups and field theory. In geometry, groups first became important in
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, ...
and, later,
non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean g ...
.
Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
's
Erlangen program In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' It is nam ...
proclaimed group theory to be the organizing principle of geometry. Galois, in the 1830s, was the first to employ groups to determine the solvability of
polynomial equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For many authors, the term '' ...
s. Arthur Cayley and
Augustin Louis Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
pushed these investigations further by creating the theory of permutation groups. The second historical source for groups stems from geometrical situations. In an attempt to come to grips with possible geometries (such as euclidean,
hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...
or
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, ...
) using group theory,
Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
initiated the
Erlangen programme In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' It is na ...
. Sophus Lie, in 1884, started using groups (now called Lie groups) attached to analytic problems. Thirdly, groups were, at first implicitly and later explicitly, used in algebraic 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 then, the impact of group theory has been ever growing, giving rise to the birth of
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
in the early 20th century,
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
, and many more influential spin-off domains. The
classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else i ...
is a vast body of work from the mid 20th century, classifying all the
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...
simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...
s.

# Main classes of groups

The range of groups being considered has gradually expanded from finite permutation groups and special examples of
matrix group In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a fa ...
s to abstract groups that may be specified through a
presentation A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
by generators and relations.

## Permutation groups

The first class of groups to undergo a systematic study was permutation groups. 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 the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
S''n''; in general, any permutation group ''G'' is a
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of the symmetric group of ''X''. An early construction due to Cayley exhibited any group as a permutation group, acting on itself () by means of the left
regular representation In mathematics, and in particular the theory of group representations, the regular representation of a group ''G'' is the linear representation afforded by the group action of ''G'' on itself by translation. One distinguishes the left regular rep ...
. 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 , the
alternating group In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic pr ...
A''n'' is
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
, i.e. does not admit any proper normal subgroups. This fact plays a key role in the impossibility of solving a general algebraic equation of degree in radicals.

## Matrix groups

The next important class of groups is given by ''matrix groups'', or
linear group In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a f ...
s. Here ''G'' is a set consisting of invertible
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
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'' by
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s. This action makes matrix groups conceptually similar to permutation groups, and the geometry of the action may be usefully exploited to establish properties of the group ''G''.

## Transformation groups

Permutation groups and matrix groups are special cases of
transformation group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the gr ...
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 a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
. The concept of a transformation group is closely related with the concept of a symmetry group: transformation groups frequently consist of ''all'' transformations that preserve a certain structure. The theory of transformation groups forms a bridge connecting group theory with differential geometry. A long line of research, originating with Lie and Klein, considers group actions on manifolds by
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
s or
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two ...
s. The groups themselves may be
discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a g ...
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 A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
by ''generators and relations'', : $G = \langle S, R\rangle.$ 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 a normal subgroup ''H''.
Class group In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a mea ...
s of algebraic number fields were among the earliest examples of factor groups, of much interest in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
. 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 In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
, as well as the classes of group with a given such property: finite groups, periodic groups,
simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...
s,
solvable group In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminate ...
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 of
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
in the works of
Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
, Emil Artin,
Emmy Noether Amalie Emmy NoetherEmmy is the '' Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noeth ...
, and mathematicians of their school.

## Groups with additional structure

An important elaboration of the concept of a group occurs if ''G'' is endowed with additional structure, notably, of a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
, differentiable manifold, or
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
. If the group operations ''m'' (multiplication) and ''i'' (inversion), : $m: G\times G\to G, \left(g,h\right)\mapsto gh, \quad i:G\to G, g\mapsto g^,$ are compatible with this structure, that is, they are continuous,
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
or regular (in the sense of algebraic geometry) maps, then ''G'' is a
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
, a Lie group, or an
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Ma ...
. 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, whereas Lie groups (frequently realized as transformation groups) are the mainstays of differential geometry and unitary
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
. 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 Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
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 groups): 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.

# Branches of group theory

## Finite group theory

During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the
local theory In physics, the principle of locality states that an object is influenced directly only by its immediate surroundings. A theory that includes the principle of locality is said to be a "local theory". This is an alternative to the concept of ins ...
of finite groups and the theory of solvable and
nilpotent group In mathematics, specifically group theory, a nilpotent group ''G'' is a group that has an upper central series that terminates with ''G''. Equivalently, its central series is of finite length or its lower central series terminates with . Intui ...
s. As a consequence, the complete
classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else i ...
was achieved, meaning that all those
simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...
s from which all finite groups can be built are now known. During the second half of the twentieth century, mathematicians such as Chevalley and
Steinberg Steinberg Media Technologies GmbH (trading as Steinberg) is a German musical software and hardware company based in Hamburg. It develops music writing, recording, arranging, and editing software, most notably Cubase, Nuendo, and Dorico. It als ...
also increased our understanding of finite analogs of
classical group In mathematics, the classical groups are defined as the special linear groups over the reals , the complex numbers and the quaternions together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or s ...
s, and other related groups. One such family of groups is the family of
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
s over
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
s. Finite groups often occur when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. The theory of Lie groups, which may be viewed as dealing with " continuous symmetry", is strongly influenced by the associated
Weyl group In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections ...
s. These are finite groups generated by reflections which act on a finite-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
. The properties of finite groups can thus play a role in subjects such as
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
and chemistry.

## Representation of groups

Saying that a group ''G'' ''
acts The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
'' on a set ''X'' means that every element of ''G'' defines a bijective map on the set ''X'' 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 In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
''V'' is a
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) w ...
: :$\rho:G \to \operatorname\left(V\right),$ where GL(''V'') consists of the invertible
linear transformations In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
of ''V''. In other words, to every group element ''g'' is assigned an automorphism ''ρ''(''g'') such that for any ''h'' in ''G''. This definition can be understood in two directions, both of which give rise to whole new domains of mathematics. 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. 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 (see Maschke's theorem). 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 it says that if ''M'' and ''N'' are two finite-dimensional irreducible representations of a group ' ...
). Given a group ''G'',
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
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 The representation theory of groups is a part of mathematics which examines how groups act on given structures. Here the focus is in particular on operations of groups on vector spaces. Nevertheless, groups acting on other groups or on sets are ...
and representations of Lie groups are two main subdomains of the theory. The totality of representations is governed by the group's
characters Character or Characters may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
. For example, Fourier polynomials can be interpreted as the characters of
U(1) In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \. ...
, the group of
complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
of absolute value ''1'', acting on the ''L''2-space of periodic functions.

## Lie theory

A Lie group is a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
that is also a differentiable manifold, with the property that the group operations are compatible with the
smooth structure In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold. Definition A smooth structure on a manifold M is ...
. Lie groups are named after Sophus Lie, who laid the foundations of the theory of continuous
transformation group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the gr ...
s. The term ''groupes de Lie'' first appeared in French in 1893 in the thesis of Lie's student
Arthur Tresse Arthur is a common male given name of Brittonic languages, Brythonic origin. Its popularity derives from it being the name of the legendary hero King Arthur. The etymology is disputed. It may derive from the Celtic ''Artos'' meaning “Bear”. An ...
, page 3. Lie groups represent the best-developed theory of continuous symmetry of
mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical p ...
s and
structures A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
. They provide a natural framework for analysing the continuous symmetries of differential equations (
differential Galois theory In mathematics, differential Galois theory studies the Galois groups of differential equations. Overview Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential field ...
), in much the same way as permutation groups are used in
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
for analysing the discrete symmetries of
algebraic equations In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
. An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations.

## 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 . A more compact way of defining a group is by ''generators and relations'', also called the ''presentation'' of a group. Given any set ''F'' of generators $\_$, the
free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
generated by ''F'' surjects onto the group ''G''. The kernel of this map is called the subgroup of relations, generated by some subset ''D''. The presentation is usually denoted by $\langle F \mid D\rangle.$ For example, the group presentation $\langle a,b\mid aba^b^\rangle$ describes a group which is isomorphic to $\mathbb\times\mathbb.$ A string consisting of generator symbols and their inverses is called a ''word''.
Combinatorial group theory In mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generators and relations. It is much used in geometric topology, the fundamental group of a simplicial complex having in a nat ...
studies groups from the perspective of generators and relations. 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
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
s via their fundamental groups. 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 A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ...
s, one can show that there is in general no
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
solving this task. Another, generally harder, algorithmically insoluble problem is the
group isomorphism problem In abstract algebra, the group isomorphism problem is the decision problem of determining whether two given finite group presentations refer to isomorphic groups. The isomorphism problem was formulated by Max Dehn, and together with the word pr ...
, which asks whether two groups given by different presentations are actually isomorphic. For example, the group with presentation $\langle x,y \mid xyxyx = e \rangle,$ is isomorphic to the additive group Z of integers, although this may not be immediately apparent. (Writing $z=xy$, one has $G \cong \langle z,y \mid z^3 = y\rangle \cong \langle z\rangle.$)
Geometric group theory 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 group (mathematics), groups and topology, topological and geometry, geometric pro ...
attacks these problems from a geometric viewpoint, either by viewing groups as geometric objects, or by finding suitable geometric objects a group acts on. The first idea is made precise by means of the
Cayley graph In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cay ...
, whose vertices correspond to group elements and edges correspond to right multiplication in the group. Given two elements, one constructs the
word metric In group theory, a word metric on a discrete group G is a way to measure distance between any two elements of G . As the name suggests, the word metric is a metric on G , assigning to any two elements g , h of G a distance d(g,h) that m ...
given by the length of the minimal path between the elements. A theorem of
Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Univ ...
and Svarc then says that given a group ''G'' acting in a reasonable manner on a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
''X'', for example a
compact manifold In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only connected one-dimensional example is ...
, then ''G'' is quasi-isometric (i.e. looks similar from a distance) to the space ''X''.

# 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 In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
map from the set to itself, giving rise to permutation groups. *If the object ''X'' is a set of points in the plane with its
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...
structure or any other
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
, a symmetry is a bijection of the set to itself which preserves the distance between each pair of points (an
isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
). The corresponding group is called
isometry group In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the ...
of ''X''. *If instead
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...
s are preserved, one speaks of conformal maps. Conformal maps give rise to
Kleinian group In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space . The latter, identifiable with , is the quotient group of the 2 by 2 complex matrices of determinant 1 by their ...
s, for example. *Symmetries are not restricted to geometrical objects, but include algebraic objects as well. For instance, the equation $x^2-3=0$ has the two solutions $\sqrt$ and $-\sqrt$. In this case, the group that exchanges the two roots is the
Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
belonging to the equation. Every polynomial equation in one variable has a Galois group, that is a certain permutation group on its roots. 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 undoing the symmetry and the associativity comes from the fact that symmetries are functions on a space, and composition of functions is associative.
Frucht's theorem Frucht's theorem is a theorem in algebraic graph theory conjectured by Dénes Kőnig in 1936 and proved by Robert Frucht in 1939. It states that every finite group is the group of symmetries of a finite undirected graph. More strongly, for any fi ...
says that every group is the symmetry group of some
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
. 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 Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
. Maps preserving the structure are then the morphisms, and the symmetry group is the
automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
of the object in question.

# Applications of group theory

Applications of group theory abound. Almost all structures in
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
are special cases of groups.
Rings Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
, for example, can be viewed as
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
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

Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
uses groups to describe the symmetries of the roots of a polynomial (or more precisely the automorphisms of the algebras generated by these roots). The fundamental theorem of Galois theory provides a link between
algebraic field extension In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, if every element of is a root of a non-zero polynomial with coefficients in . A field ext ...
s and group theory. It gives an effective criterion for the solvability of polynomial equations in terms of the solvability of the corresponding
Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
. For example, ''S''5, the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
in 5 elements, is not solvable which implies that the general
quintic equation In algebra, a quintic function is a function of the form :g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\, where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a ...
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 as class field theory.

## Algebraic topology

Algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
is another domain which prominently associates groups to the objects the theory is interested in. There, groups are used to describe certain invariants of
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
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 Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies. * Defor ...
. For example, the fundamental group "counts" how many paths in the space are essentially different. The
Poincaré conjecture In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured ...
, proved in 2002/2003 by
Grigori Perelman Grigori Yakovlevich Perelman ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian mathemati ...
, is a prominent application of this idea. The influence is not unidirectional, though. For example, algebraic topology makes use of
Eilenberg–MacLane space In mathematics, specifically algebraic topology, an Eilenberg–MacLane space Saunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name ...
s which are spaces with prescribed
homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...
. Similarly
algebraic K-theory Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense ...
relies in a way on
classifying space In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free ac ...
s of groups. Finally, the name of the torsion subgroup of an infinite group shows the legacy of topology in group theory.

## Algebraic geometry

Algebraic geometry likewise uses group theory in many ways.
Abelian varieties In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group law th ...
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
Hodge conjecture In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties. In simple terms, the Hodge conjectu ...
(in certain cases).) The one-dimensional case, namely
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
s is studied in particular detail. They are both theoretically and practically intriguing. In another direction, toric varieties are
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
acted on by a
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...
. Toroidal embeddings have recently led to advances in algebraic geometry, in particular resolution of singularities.

## Algebraic number theory

Algebraic number theory makes uses of groups for some important applications. For example, Euler's product formula, :$\begin \sum_\frac& = \prod_ \frac, \\ \end \!$ captures the fact that any integer decomposes in a unique way into
primes A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
. The failure of this statement for more general rings gives rise to
class group In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a mea ...
s and
regular prime In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli nu ...
s, which feature in Kummer's treatment of
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been ...
.

## Harmonic analysis

Analysis on Lie groups and certain other groups is called harmonic analysis. Haar measures, that is, integrals invariant under the translation in a Lie group, are used for
pattern recognition Pattern recognition is the automated recognition of patterns and regularities in data. It has applications in statistical data analysis, signal processing, image analysis, information retrieval, bioinformatics, data compression, computer graphics ...
and other image processing techniques.

## Combinatorics

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.

## Music

The presence of the 12- periodicity in the
circle of fifths In music theory, the circle of fifths is a way of organizing the 12 chromatic pitches as a sequence of perfect fifths. (This is strictly true in the standard 12-tone equal temperament system — using a different system requires one interval of ...
yields applications of elementary group theory in
musical set theory Musical set theory provides concepts for categorizing musical objects and describing their relationships. Howard Hanson first elaborated many of the concepts for analyzing tonal music. Other theorists, such as Allen Forte, further developed th ...
. Transformational theory models musical transformations as elements of a mathematical group.

## Physics

In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, groups are important because they describe the symmetries which the laws of physics seem to obey. According to
Noether's theorem Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
, every continuous symmetry of a physical system corresponds to a conservation law of the system. Physicists are very interested in group representations, especially of Lie groups, since these representations often point the way to the "possible" physical theories. Examples of the use of groups in physics include the Standard Model, gauge theory, the
Lorentz group In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
, and the
Poincaré group The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
. Group theory can be used to resolve the incompleteness of the statistical interpretations of mechanics developed by
Willard Gibbs Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in ...
, relating to the summing of an infinite number of probabilities to yield a meaningful solution. Norbert Wiener, Cybernetics: Or Control and Communication in the Animal and the Machine, , Ch 2

## Chemistry and materials science

In chemistry and materials science, point groups are used to classify regular polyhedra, and the symmetries of molecules, and
space group In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it uncha ...
s to classify
crystal structure In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric patterns ...
s. The assigned groups can then be used to determine physical properties (such as chemical polarity and chirality), spectroscopic properties (particularly useful for Raman spectroscopy,
infrared spectroscopy Infrared spectroscopy (IR spectroscopy or vibrational spectroscopy) is the measurement of the interaction of infrared radiation with matter by absorption, emission, or reflection. It is used to study and identify chemical substances or function ...
, circular dichroism spectroscopy, magnetic circular dichroism spectroscopy, UV/Vis spectroscopy, and fluorescence spectroscopy), and to construct
molecular orbital In chemistry, a molecular orbital is a mathematical function describing the location and wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of findin ...
s. Molecular symmetry is responsible for many physical and spectroscopic properties of compounds and provides relevant information about how chemical reactions occur. In order to assign a point group for any given molecule, it is necessary to find the set of symmetry operations present on it. The symmetry operation is an action, such as a rotation around an axis or a reflection through a mirror plane. In other words, it is an operation that moves the molecule such that it is indistinguishable from the original configuration. In group theory, the rotation axes and mirror planes are called "symmetry elements". These elements can be a point, line or plane with respect to which the symmetry operation is carried out. The symmetry operations of a molecule determine the specific point group for this molecule. In chemistry, there are five important symmetry operations. They are identity operation (E), rotation operation or proper rotation (C''n''), reflection operation (σ), inversion (i) and rotation reflection operation or improper rotation (S''n''). The identity operation (E) consists of leaving the molecule as it is. This is equivalent to any number of full rotations around any axis. This is a symmetry of all molecules, whereas the symmetry group of a
chiral Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from i ...
molecule consists of only the identity operation. An identity operation is a characteristic of every molecule even if it has no symmetry. Rotation around an axis (C''n'') consists of rotating the molecule around a specific axis by a specific angle. It is rotation through the angle 360°/''n'', where ''n'' is an integer, about a rotation axis. For example, if a
water Water (chemical formula ) is an Inorganic compound, inorganic, transparent, tasteless, odorless, and Color of water, nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living ...
molecule rotates 180° around the axis that passes through the
oxygen Oxygen is the chemical element with the symbol O and atomic number 8. It is a member of the chalcogen group in the periodic table, a highly reactive nonmetal, and an oxidizing agent that readily forms oxides with most elements as ...
atom and between the
hydrogen Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic ...
atoms, it is in the same configuration as it started. In this case, , since applying it twice produces the identity operation. In molecules with more than one rotation axis, the Cn axis having the largest value of n is the highest order rotation axis or principal axis. For example in boron trifluoride (BF3), the highest order of rotation axis is C3, so the principal axis of rotation is C3. In the reflection operation (σ) many molecules have mirror planes, although they may not be obvious. The reflection operation exchanges left and right, as if each point had moved perpendicularly through the plane to a position exactly as far from the plane as when it started. When the plane is perpendicular to the principal axis of rotation, it is called σ''h'' (horizontal). Other planes, which contain the principal axis of rotation, are labeled vertical (σ''v'') or dihedral (σ''d''). Inversion (i ) is a more complex operation. Each point moves through the center of the molecule to a position opposite the original position and as far from the central point as where it started. Many molecules that seem at first glance to have an inversion center do not; for example,
methane Methane ( , ) is a chemical compound with the chemical formula (one carbon atom bonded to four hydrogen atoms). It is a group-14 hydride, the simplest alkane, and the main constituent of natural gas. The relative abundance of methane on Ea ...
and other
tetrahedral In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...
molecules lack inversion symmetry. To see this, hold a methane model with two hydrogen atoms in the vertical plane on the right and two hydrogen atoms in the horizontal plane on the left. Inversion results in two hydrogen atoms in the horizontal plane on the right and two hydrogen atoms in the vertical plane on the left. Inversion is therefore not a symmetry operation of methane, because the orientation of the molecule following the inversion operation differs from the original orientation. And the last operation is improper rotation or rotation reflection operation (S''n'') requires rotation of  360°/''n'', followed by reflection through a plane perpendicular to the axis of rotation.

## Cryptography

Very large groups of prime order constructed in elliptic curve cryptography serve for
public-key cryptography Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic alg ...
. 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. One of the earliest encryption protocols, Caesar's cipher, may also be interpreted as a (very easy) group operation. Most cryptographic schemes use groups in some way. In particular Diffie–Hellman key exchange uses finite cyclic groups. So the term group-based cryptography refers mostly to cryptographic protocols that use infinite nonabelian groups such as a braid group.

*
List of group theory topics In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all ...
*
Examples of groups Some elementary examples of groups in mathematics are given on Group (mathematics). Further examples are listed here. Permutations of a set of three elements Consider three colored blocks (red, green, and blue), initially placed in the order R ...

# References

* * * * * Shows the advantage of generalising from group to
groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: *'' Group'' with a partial func ...
. * An introductory undergraduate text in the spirit of texts by Gallian or Herstein, covering groups, rings, integral domains, fields and Galois theory. Free downloadable PDF with open-source
GFDL The GNU Free Documentation License (GNU FDL or simply GFDL) is a copyleft license for free documentation, designed by the Free Software Foundation (FSF) for the GNU Project. It is similar to the GNU General Public License, giving readers the ...
license. * * * Conveys the practical value of group theory by explaining how it points to symmetries in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and other sciences. * * Ronan M., 2006. ''Symmetry and the Monster''. Oxford University Press. . For lay readers. Describes the quest to find the basic building blocks for finite groups. * A standard contemporary reference. * * Inexpensive and fairly readable, but somewhat dated in emphasis, style, and notation. * *