TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
and
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
, group theory studies the
algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s known as
groups A group is a number of people 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 identi ...
. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as
rings Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to: *Ring (jewellery) A ring is a round band, usually of metal A metal (from Ancient Greek, Greek μέταλλον ''métallon'', "mine ...
,
fields File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe'' FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...
, and
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s, can all be seen as groups endowed with additional operations and
axiom An axiom, postulate or assumption is a statement that is taken to be , to serve as a or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or 'that which comm ...

s. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra.
Linear algebraic group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s and
Lie group In mathematics, a Lie group (pronounced "Lee") is a group (mathematics), group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operati ...
s 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, macrosco ...

s and the
hydrogen atom A hydrogen atom is an atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touch ...

, may be modelled by
symmetry group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 ...
s. Thus group theory and the closely related
representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
have many important applications in
physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

,
chemistry Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo during a with other . ...

, and
materials science The interdisciplinary Interdisciplinarity or interdisciplinary studies involves the combination of two or more academic discipline An academic discipline or academic field is a subdivision of knowledge that is Education, taught and resea ...
. Group theory is also central to
public key cryptography 250px, In this example the message is digitally signed, but not encrypted. 1) Alice signs a message with her private key. 2) Bob can verify that Alice sent the message and that the message has not been modified. Public-key cryptography, or ...
. The early
history of group theoryThe history of group theory, a mathematics, mathematical domain studying group (mathematics), groups in their various forms, has evolved in various parallel threads. There are three historical roots of group theory: the theory of algebraic equations, ...
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 1980, that culminated in a complete
classification of finite simple groups In mathematics, the classification of the finite simple groups is a theorem stating that every List of finite simple groups, finite simple group is either cyclic groups, cyclic, or alternating groups, alternating, or it belongs to a broad infinite ...
.

# Main classes of groups

The range of groups being considered has gradually expanded from
finite Finite is the opposite of Infinity, 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 ...
permutation group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s and special examples of
matrix groupIn mathematics, a matrix group is a group (mathematics), group ''G'' consisting of invertible matrix, invertible matrix (mathematics), matrices over a specified field (mathematics), field ''K'', with the operation of matrix multiplication. A linear g ...
s to abstract groups that may be specified through a
presentation A presentation conveys information from a speaker to an audience An audience is a group of people who participate in a show or encounter a work of art A work of art, artwork, art piece, piece of art or art object is an a ...
by generators and
relations Relation or relations may refer to: General uses *International relations, the study of interconnection of politics, economics, and law on a global level *Interpersonal relationship, association or acquaintance between two or more people *Public ...
.

## Permutation groups

The first
class Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differently f ...
of groups to undergo a systematic study was
permutation group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s. Given any set ''X'' and a collection ''G'' of
bijection In , a bijection, bijective function, one-to-one correspondence, or invertible function, is a between the elements of two , where each element of one set is paired with exactly one element of the other set, and each element of the other set is p ...

s of ''X'' into itself (known as ''permutations'') that is closed under compositions and inverses, ''G'' is a group
acting Acting is an activity in which a story is told by means of its enactment Enactment may refer to: Law * Enactment of a bill, when a bill becomes law * Enacting formula, formulaic words in a bill or act which introduce its provisions * Enactm ...
on ''X''. If ''X'' consists of ''n'' elements and ''G'' consists of ''all'' permutations, ''G'' is the
symmetric group In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematic ...
S''n''; in general, any permutation group ''G'' is a
subgroup In group theory, a branch of mathematics, given a group (mathematics), 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 ...
of the symmetric group of ''X''. An early construction due to exhibited any group as a permutation group, acting on itself () by means of the left
regular representation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
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 subgroup In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
s. 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
s. Here ''G'' is a set consisting of invertible
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
of given order ''n'' over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
''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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
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 (mathematics), group consisting of automorphisms of ''X''. For example, if ''X'' is a Dimension (vector space), finite-dimensional vector space, then the automorphism group of ' ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
. The concept of a transformation group is closely related with the concept of a
symmetry group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 ...
: 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 Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
. A long line of research, originating with
Lie A lie is an assertion that is believed to be false, typically used with the purpose of deception, deceiving someone. The practice of communicating lies is called lying. A person who communicates a lie may be termed a liar. Lies may serve a vari ...

and Klein, considers group actions on
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

s by
homeomorphism In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantiti ...
s or
diffeomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s. The groups themselves may be
discrete Discrete in science is the opposite of :wikt:continuous, continuous: something that is separate; distinct; individual. Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic c ...
or
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
.

## 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 An audience is a group of people who participate in a show or encounter a work of art A work of art, artwork, art piece, piece of art or art object is an a ...
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 A quotient group or factor group is a mathematical group (mathematics), group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factore ...
, ''G''/''H'', of a group ''G'' by a
normal subgroup In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
''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 measu ...
s of
algebraic number field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s 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 devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

. 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, as well as the classes of group with a given such property:
finite group In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
s, periodic groups,
simple group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
s,
solvable group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
in the works of
Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in man ...
,
Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as qua ...
,
Emmy Noether Amalie Emmy Noether Emmy 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.

An important elaboration of the concept of a group occurs if ''G'' is endowed with additional structure, notably, of a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
,
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surfa ...
, or
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematic ...
. 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 Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
,
smooth Smooth may refer to: Mathematics * Smooth function is a smooth function with compact support. In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuo ...
or
regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * Regular (Badfinger song), "Regular" (Badfinger song) * Regular tunin ...
(in the sense of algebraic geometry) maps, then ''G'' is a
topological group In mathematics, topological groups are logically the combination of Group (mathematics), groups and Topological space, topological spaces, i.e. they are groups and topological spaces at the same time, such that the Continuous function, continui ...
, a
Lie group In mathematics, a Lie group (pronounced "Lee") is a group (mathematics), group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operati ...
, or an
algebraic group In algebraic geometry, an algebraic group (or group variety) is a Group (mathematics), group that is an algebraic variety, such that the multiplication and inversion operations are given by regular map (algebraic geometry), regular maps on the varie ...
. 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 group In mathematics, a Lie group (pronounced "Lee") is a group (mathematics), group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operati ...
s (frequently realized as transformation groups) are the mainstays of
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
and unitary
representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
. 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 groupIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
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.

# 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 Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
of finite groups and the theory of solvable and
nilpotent group In mathematics, specifically group theory, a nilpotent group ''G'' is a Group (mathematics), group that has an upper central series that terminates with ''G''. Equivalently, its central series is of finite length or its lower central series termina ...
s. As a consequence, the complete
classification of finite simple groups In mathematics, the classification of the finite simple groups is a theorem stating that every List of finite simple groups, finite simple group is either cyclic groups, cyclic, or alternating groups, alternating, or it belongs to a broad infinite ...
was achieved, meaning that all those
simple group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
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 with satellite offices in Siegburg and London. It develops music writing, recording, arranging, and editing software, mos ...
also increased our understanding of finite analogs of
classical groups In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, and other related groups. One such family of groups is the family of
general linear group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s over
finite field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s. Finite groups often occur when considering
symmetry Symmetry (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ...

of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. The theory of
Lie group In mathematics, a Lie group (pronounced "Lee") is a group (mathematics), group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operati ...
s, which may be viewed as dealing with "
continuous symmetry In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some Symmetry in mathematics, symmetries as Motion (physics), motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant un ...
", is strongly influenced by the associated
Weyl group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s. These are finite groups generated by reflections which act on a finite-dimensional
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
. 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 List of natural phenomena, natural phenomena. This is in contrast to experimental ph ...
and
chemistry Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo during a with other . ...

.

## Representation of groups

Saying that a group ''G'' ''
acts The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum), often referred to simply as Acts, or formally the Book of Acts, is the fifth book of the New Testament The New Te ...
'' 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
''V'' is a
group homomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

: :$\rho:G \to \operatorname\left(V\right),$ where GL(''V'') consists of the invertible
linear transformations In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of ''V''. In other words, to every group element ''g'' is assigned an
automorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

''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 , 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. These parts in turn are much more easily manageable than the whole ''V'' (via Schur's lemma). Given a group ''G'',
representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
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 group (mathematics), groups is a part of mathematics which examines how groups act on given structures. Here the focus is in particular on Group action (mathematics), operations of groups on vector spaces. Nevertheless, ...
and representations of
Lie group In mathematics, a Lie group (pronounced "Lee") is a group (mathematics), group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operati ...
s are two main subdomains of the theory. The totality of representations is governed by the group's
characters Character(s) may refer to: Arts, entertainment, and media Literature * Character (novel), ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * Characters (Theophrastus), ''Characters'' (Theophrastus), a classical Greek set of char ...
. For example, Fourier series, Fourier polynomials can be interpreted as the characters of unitary group, U(1), the group of complex numbers of absolute value ''1'', acting on the Lp space, ''L''2-space of periodic functions.

## Lie theory

A
Lie group In mathematics, a Lie group (pronounced "Lee") is a group (mathematics), group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operati ...
is a group (mathematics), group that is also a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surfa ...
, with the property that the group operations are compatible with the Differential structure, smooth structure. 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 (mathematics), group consisting of automorphisms of ''X''. For example, if ''X'' is a Dimension (vector space), finite-dimensional vector space, then the automorphism group of ' ...
s. The term ''groupes de Lie'' first appeared in French in 1893 in the thesis of Lie's student Arthur Tresse, page 3. Lie groups represent the best-developed theory of
continuous symmetry In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some Symmetry in mathematics, symmetries as Motion (physics), motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant un ...
of mathematical objects and mathematical structure, structures, 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 List of natural phenomena, natural phenomena. This is in contrast to experimental ph ...
. They provide a natural framework for analysing the continuous symmetries of differential equations (differential Galois theory), in much the same way as permutation groups are used in Galois theory for analysing the discrete symmetries of algebraic equations. 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 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 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 (discrete mathematics), graphs 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 for groups, word problem'' asks whether two words are effectively the same group element. By relating the problem to Turing machines, one can show that there is in general no algorithm solving this task. Another, generally harder, algorithmically insoluble problem is the group isomorphism problem, 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. 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. The first idea is made precise by means of the Cayley graph, whose vertices correspond to group elements and edges correspond to right multiplication in the group. Given two elements, one constructs the word metric given by the length of the minimal path between the elements. A theorem of John Milnor, Milnor and Svarc then says that given a group ''G'' acting in a reasonable manner on a metric space ''X'', for example a compact manifold, then ''G'' is quasi-isometry, 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 Symmetry (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ...

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 bijection, bijective map from the set to itself, giving rise to
permutation group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s. #If the object ''X'' is a set of points in the plane with its metric (mathematics), metric structure or any other metric space, a symmetry is a
bijection In , a bijection, bijective function, one-to-one correspondence, or invertible function, is a between the elements of two , where each element of one set is paired with exactly one element of the other set, and each element of the other set is p ...

of the set to itself which preserves the distance between each pair of points (an isometry). The corresponding group is called isometry group of ''X''. #If instead angles are preserved, one speaks of conformal maps. Conformal maps give rise to Kleinian groups, 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 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 Symmetry (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ...

. Symmetries form a group: they are closure (mathematics), 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 says that every group is the symmetry group of some Graph (discrete mathematics), 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 (mathematics), category. Maps preserving the structure are then the morphisms, and the symmetry group is the automorphism group of the object in question.

# Applications of group theory

Applications of group theory abound. Almost all structures in
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
are special cases of groups. Ring (mathematics), Rings, for example, can be viewed as abelian groups (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 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 extensions and group theory. It gives an effective criterion for the solvability of polynomial equations in terms of the solvability of the corresponding Galois group. For example, ''S''5, the
symmetric group In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematic ...
in 5 elements, is not solvable which implies that the general quintic equation cannot be solved by radicals in the way equations of lower degree can. 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 is another domain which prominently functor, associates groups to the objects the theory is interested in. There, groups are used to describe certain invariants of
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
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 homeomorphism, deformation. For example, the fundamental group "counts" how many paths in the space are essentially different. The Poincaré conjecture, proved in 2002/2003 by Grigori Perelman, is a prominent application of this idea. The influence is not unidirectional, though. For example, algebraic topology makes use of Eilenberg–MacLane spaces which are spaces with prescribed homotopy groups. Similarly algebraic K-theory relies in a way on classifying spaces 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 variety, 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. The one-dimensional case, namely elliptic curves is studied in particular detail. They are both theoretically and practically intriguing. In another direction, toric variety, toric varieties are algebraic variety, algebraic varieties acted on by a torus. 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 product, Euler's product formula, :$\begin \sum_\frac& = \prod_ \frac, \\ \end \!$ captures Fundamental theorem of arithmetic, the fact that any integer decomposes in a unique way into prime number, primes. The failure of this statement for Dedekind ring, more general rings gives rise to class groups and regular primes, which feature in Ernst Kummer, Kummer's treatment of Fermat's Last Theorem.

## 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 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-Periodic group, periodicity in the circle of fifths yields applications of elementary group theory in set theory (music), musical set theory. Transformational theory models musical transformations as elements of a mathematical group.

## Physics

In
physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

, groups are important because they describe the symmetries which the laws of physics seem to obey. According to Noether's theorem, every continuous symmetry of a physical system corresponds to a Conservation law (physics), 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, and the Poincaré group. Group theory can be used to resolve the incompleteness of the statistical interpretations of mechanics developed by Josiah Willard Gibbs, Willard Gibbs, 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 Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo during a with other . ...

and
materials science The interdisciplinary Interdisciplinarity or interdisciplinary studies involves the combination of two or more academic discipline An academic discipline or academic field is a subdivision of knowledge that is Education, taught and resea ...
, point groups are used to classify regular polyhedra, and the molecular symmetry, symmetries of molecules, and space groups to classify crystal structures. The assigned groups can then be used to determine physical properties (such as chemical polarity and Chirality (chemistry), chirality), spectroscopic properties (particularly useful for Raman spectroscopy, infrared spectroscopy, circular dichroism spectroscopy, magnetic circular dichroism spectroscopy, UV/Vis spectroscopy, and fluorescence spectroscopy), and to construct molecular orbitals. 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 Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo during a with other . ...

, 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 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 molecule rotates 180° around the axis that passes through the oxygen atom and between the hydrogen 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 and other Tetrahedron, tetrahedral 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. 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 cipher, 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.

# History

Group theory has three main historical sources:
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

, the theory of algebraic equations, and geometry. The number-theoretic strand was begun by Leonhard Euler, and developed by Carl Friedrich Gauss, Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields. Early results about
permutation group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s were obtained by Joseph Louis Lagrange, Lagrange, Paolo Ruffini, Ruffini, and Niels Henrik Abel, Abel in their quest for general solutions of polynomial equations of high degree. Évariste Galois coined the term "group" and established a connection, now known as Galois theory, between the nascent theory of groups and field theory (mathematics), field theory. In geometry, groups first became important in projective geometry and, later, non-Euclidean geometry. Felix Klein's Erlangen program proclaimed group theory to be the organizing principle of geometry. Évariste Galois, Galois, in the 1830s, was the first to employ groups to determine the solvability of polynomial equations. Arthur Cayley and Augustin Louis Cauchy pushed these investigations further by creating the theory of permutation groups. The second historical source for groups stems from geometry, geometrical situations. In an attempt to come to grips with possible geometries (such as euclidean geometry, euclidean, hyperbolic geometry, hyperbolic or projective geometry) using group theory, Felix Klein initiated the Erlangen programme. Sophus Lie, in 1884, started using groups (now called
Lie group In mathematics, a Lie group (pronounced "Lee") is a group (mathematics), group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operati ...
s) attached to analysis (mathematics), 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 algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
in the early 20th century,
representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
, and many more influential spin-off domains. The
classification of finite simple groups In mathematics, the classification of the finite simple groups is a theorem stating that every List of finite simple groups, finite simple group is either cyclic groups, cyclic, or alternating groups, alternating, or it belongs to a broad infinite ...
is a vast body of work from the mid 20th century, classifying all the finite set, finite
simple group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
s.

*List of group theory topics *Examples of groups

# References

* * * * * Shows the advantage of generalising from group to groupoid. * 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 GNU Free Documentation License, GFDL license. * * * Conveys the practical value of group theory by explaining how it points to symmetries in
physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

and other sciences. * * Mark Ronan, 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. * *