In
mathematics, a group is a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
and an
operation
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Ma ...
that combines any two
elements of the set to produce a third element of the set, in such a way that the operation is
associative, an
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
exists and every element has an
inverse. These three
axioms hold for
number systems
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
and many other mathematical structures. For example, the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers,
geometric shapes and
polynomial root
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or equ ...
s. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics.
In
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 ...
groups arise naturally in the study of
symmetries
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
and
geometric transformation
In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and range are sets of points — most often b ...
s: The symmetries of an object form a group, called the
symmetry group of the object, and the transformations of a given type form a general group.
Lie groups appear in symmetry groups in geometry, and also in the
Standard Model of
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
. 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 ...
is a Lie group consisting of the symmetries of
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
in
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws ...
.
Point groups describe
symmetry in molecular chemistry.
The concept of a group arose in the study 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, starting with
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, ...
in the 1830s, who introduced the term ''group'' (French: ) for the symmetry group of the
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
of an equation, now called a
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 ...
. After contributions from other fields such as
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 ...
and geometry, the group notion was generalized and firmly established around 1870. Modern
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as
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 ...
s,
quotient groups and
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. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely, both from a point of view of
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 ...
(that is, through the
representations of the group) and of
computational group theory
In mathematics, computational group theory is the study of
groups by means of computers. It is concerned
with designing and analysing algorithms and
data structures to compute information about groups. The subject
has attracted interest because f ...
. A theory has been developed for
finite groups, which culminated with 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 ...
, completed in 2004. Since the mid-1980s,
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 ...
, which studies
finitely generated group
In algebra, a finitely generated group is a group ''G'' that has some finite generating set ''S'' so that every element of ''G'' can be written as the combination (under the group operation) of finitely many elements of ''S'' and of inverses o ...
s as geometric objects, has become an active area in group theory.
Definition and illustration
First example: the integers
One of the more familiar groups is the set of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s
together with
addition. For any two integers
and
, the
sum is also an integer; this ''
closure'' property says that
is a
binary operation on
. The following properties of integer addition serve as a model for the group axioms in the definition below.
*For all integers
,
and
, one has
. Expressed in words, adding
to
first, and then adding the result to
gives the same final result as adding
to the sum of
and
. This property is known as ''
associativity
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
''.
*If
is any integer, then
and
.
Zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
is called the ''
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
'' of addition because adding it to any integer returns the same integer.
*For every integer
, there is an integer
such that
and
. The integer
is called the ''
inverse element
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
'' of the integer
and is denoted
.
The integers, together with the operation
, form a mathematical object belonging to a broad class sharing similar structural aspects. To appropriately understand these structures as a collective, the following definition is developed.
Definition
A group is a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
together with a
binary operation on
, here denoted "
", that combines any two
elements and
to form an element of
, denoted
, such that the following three requirements, known as ''group axioms'', are satisfied:
;Associativity: For all
,
,
in
, one has
.
;Identity element: There exists an element
in
such that, for every
in
, one has
and
.
:Such an element is unique (
see below). It is called ''the identity element'' of the group.
;Inverse element: For each
in
, there exists an element
in
such that
and
, where
is the identity element.
:For each
, the element
is unique (
see below); it is called ''the inverse'' of
and is commonly denoted
.
Notation and terminology
Formally, the group is the
ordered pair of a set and a binary operation on this set that satisfies the
group axioms
In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. Thes ...
. The set is called the ''underlying set'' of the group, and the operation is called the ''group operation'' or the ''group law''.
A group and its underlying set are thus two different
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. To avoid cumbersome notation, it is common to
abuse notation by using the same symbol to denote both. This reflects also an informal way of thinking: that the group is the same as the set except that it has been enriched by additional structure provided by the operation.
For example, consider the set of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s
, which has the operations of addition
and
multiplication . Formally,
is a set,
is a group, and
is 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 grass ...
. But it is common to write
to denote any of these three objects.
The ''additive group'' of the field
is the group whose underlying set is
and whose operation is addition. The ''multiplicative group'' of the field
is the group
whose underlying set is the set of nonzero real numbers
and whose operation is multiplication.
More generally, one speaks of an ''additive group'' whenever the group operation is notated as addition; in this case, the identity is typically denoted
, and the inverse of an element
is denoted
. Similarly, one speaks of a ''multiplicative group'' whenever the group operation is notated as multiplication; in this case, the identity is typically denoted
, and the inverse of an element
is denoted
. In a multiplicative group, the operation symbol is usually omitted entirely, so that the operation is denoted by juxtaposition,
instead of
.
The definition of a group does not require that
for all elements
and
in
. If this additional condition holds, then the operation is said to be
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
, and the group is called an
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 ...
. It is a common convention that for an abelian group either additive or multiplicative notation may be used, but for a nonabelian group only multiplicative notation is used.
Several other notations are commonly used for groups whose elements are not numbers. For a group whose elements are
functions, the operation is often
function composition ; then the identity may be denoted id. In the more specific cases of
geometric transformation
In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and range are sets of points — most often b ...
groups,
symmetry groups,
permutation groups, and
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 ...
s, the symbol
is often omitted, as for multiplicative groups. Many other variants of notation may be encountered.
Second example: a symmetry group
Two figures in the
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
are
congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
if one can be changed into the other using a combination of
rotations,
reflection Reflection or reflexion may refer to:
Science and technology
* Reflection (physics), a common wave phenomenon
** Specular reflection, reflection from a smooth surface
*** Mirror image, a reflection in a mirror or in water
** Signal reflection, in ...
s, and
translation
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
s. Any figure is congruent to itself. However, some figures are congruent to themselves in more than one way, and these extra congruences are called
symmetries
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
. A
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
has eight symmetries. These are:
* the
identity operation leaving everything unchanged, denoted id;
* rotations of the square around its center by 90°, 180°, and 270° clockwise, denoted by
,
and
, respectively;
* reflections about the horizontal and vertical middle line (
and
), or through the two
diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
s (
and
).
These symmetries are functions. Each sends a point in the square to the corresponding point under the symmetry. For example,
sends a point to its rotation 90° clockwise around the square's center, and
sends a point to its reflection across the square's vertical middle line. Composing two of these symmetries gives another symmetry. These symmetries determine a group called the
dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ...
of degree four, denoted
. The underlying set of the group is the above set of symmetries, and the group operation is function composition. Two symmetries are combined by composing them as functions, that is, applying the first one to the square, and the second one to the result of the first application. The result of performing first
and then
is written symbolically ''from right to left'' as
("apply the symmetry
after performing the symmetry
"). This is the usual notation for composition of functions.
The
group table Named after the 19th century United Kingdom, British mathematician Arthur Cayley, a Cayley table describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an additio ...
lists the results of all such compositions possible. For example, rotating by 270° clockwise (
) and then reflecting horizontally (
) is the same as performing a reflection along the diagonal (
). Using the above symbols, highlighted in blue in the group table:
Given this set of symmetries and the described operation, the group axioms can be understood as follows.
''Binary operation'': Composition is a binary operation. That is,
is a symmetry for any two symmetries
and
. For example,
that is, rotating 270° clockwise after reflecting horizontally equals reflecting along the counter-diagonal (
). Indeed, every other combination of two symmetries still gives a symmetry, as can be checked using the group table.
''Associativity'': The associativity axiom deals with composing more than two symmetries: Starting with three elements
,
and
of
, there are two possible ways of using these three symmetries in this order to determine a symmetry of the square. One of these ways is to first compose
and
into a single symmetry, then to compose that symmetry with
. The other way is to first compose
and
, then to compose the resulting symmetry with
. These two ways must give always the same result, that is,
For example,
can be checked using the group table:
''Identity element'': The identity element is
, as it does not change any symmetry
when composed with it either on the left or on the right.
''Inverse element'': Each symmetry has an inverse:
, the reflections
,
,
,
and the 180° rotation
are their own inverse, because performing them twice brings the square back to its original orientation. The rotations
and
are each other's inverses, because rotating 90° and then rotation 270° (or vice versa) yields a rotation over 360° which leaves the square unchanged. This is easily verified on the table.
In contrast to the group of integers above, where the order of the operation is immaterial, it does matter in
, as, for example,
but
. In other words,
is not abelian.
History
The modern concept of an
abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions 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 of degree higher than 4. The 19th-century French mathematician
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, ...
, extending prior work of
Paolo Ruffini
Paolo Ruffini (Valentano, 22 September 1765 – Modena, 10 May 1822) was an Italian mathematician and philosopher.
Education and Career
By 1788 he had earned university degrees in philosophy, medicine/surgery and mathematics. His works inclu ...
and
Joseph-Louis Lagrange, gave a criterion for the
solvability of a particular polynomial equation in terms of the
symmetry group of its
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
(solutions). The elements of such a
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 ...
correspond to certain
permutations of the roots. At first, Galois's ideas were rejected by his contemporaries, and published only posthumously. More general permutation groups were investigated in particular by
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 ...
.
Arthur Cayley's ''On the theory of groups, as depending on the symbolic equation
'' (1854) gives the first abstract definition of a
finite group.
Geometry was a second field in which groups were used systematically, especially symmetry groups as part of
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 1872
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 ...
. After novel geometries such as
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 ...
and
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, ...
had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas,
Sophus Lie
Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations.
Life and career
Marius Soph ...
founded the study of
Lie groups in 1884.
The third field contributing to group theory was
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 ...
. Certain abelian group structures had been used implicitly in
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
's number-theoretical work ''
Disquisitiones Arithmeticae
The (Latin for "Arithmetical Investigations") is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. It is notable for having had a revolutionary impact on th ...
'' (1798), and more explicitly by
Leopold Kronecker
Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by as having said, "'" ("God made the integers, ...
. In 1847,
Ernst Kummer
Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned ...
made early attempts to prove
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 ...
by developing
groups describing factorization into
prime number
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 ...
s.
The convergence of these various sources into a uniform theory of groups started with
Camille Jordan
Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''.
Biography
Jordan was born in Lyon and educated at ...
's (1870).
Walther von Dyck
Walther Franz Anton von Dyck (6 December 1856 – 5 November 1934), born Dyck () and later ennobled, was a German mathematician. He is credited with being the first to define a mathematical group, in the modern sense in . He laid the foundations ...
(1882) introduced the idea of specifying a group by means of generators and relations, and was also the first to give an axiomatic definition of an "abstract group", in the terminology of the time. As of the 20th century, groups gained wide recognition by the pioneering work of
Ferdinand Georg Frobenius and
William Burnside
:''This English mathematician is sometimes confused with the Irish mathematician William S. Burnside (1839–1920).''
__NOTOC__
William Burnside (2 July 1852 – 21 August 1927) was an English mathematician. He is known mostly as an early rese ...
, who worked on
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 ...
of finite groups,
Richard Brauer
Richard Dagobert Brauer (February 10, 1901 – April 17, 1977) was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory. He was the founder of modular represent ...
's
modular representation theory
Modular representation theory is a branch of mathematics, and is the part of representation theory that studies linear representations of finite groups over a field ''K'' of positive characteristic ''p'', necessarily a prime number. As well as ...
and
Issai Schur
Issai Schur (10 January 1875 – 10 January 1941) was a Russian mathematician who worked in Germany for most of his life. He studied at the University of Berlin. He obtained his doctorate in 1901, became lecturer in 1903 and, after a stay at th ...
's papers. The theory of Lie groups, and more generally
locally compact group
In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are loc ...
s was studied by
Hermann Weyl,
Élie Cartan
Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...
and many others. Its
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
ic counterpart, the theory of
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 ...
s, was first shaped by
Claude Chevalley
Claude Chevalley (; 11 February 1909 – 28 June 1984) was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory and the theory of algebraic groups. He was a foun ...
(from the late 1930s) and later by the work of
Armand Borel
Armand Borel (21 May 1923 – 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in alg ...
and
Jacques Tits
Jacques Tits () (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric.
Life an ...
.
The
University of Chicago
The University of Chicago (UChicago, Chicago, U of C, or UChi) is a private university, private research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park, Chicago, Hyde Park neighborhood. The University of Chic ...
's 1960–61 Group Theory Year brought together group theorists such as
Daniel Gorenstein
Daniel E. Gorenstein (January 1, 1923 – August 26, 1992) was an American mathematician. He earned his undergraduate and graduate degrees at Harvard University, where he earned his Ph.D. in 1950 under Oscar Zariski, introducing in his dissertati ...
,
John G. Thompson
John Griggs Thompson (born October 13, 1932) is an American mathematician at the University of Florida noted for his work in the field of finite groups. He was awarded the Fields Medal in 1970, the Wolf Prize in 1992, and the Abel Prize in 2008.
...
and
Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to 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 ...
, with the final step taken by
Aschbacher and Smith in 2004. This project exceeded previous mathematical endeavours by its sheer size, in both length of
proof and number of researchers. Research concerning this classification proof is ongoing. Group theory remains a highly active mathematical branch, impacting many other fields, as the
examples below illustrate.
Elementary consequences of the group axioms
Basic facts about all groups that can be obtained directly from the group axioms are commonly subsumed under ''elementary group theory''. For example,
repeated applications of the associativity axiom show that the unambiguity of
generalizes to more than three factors. Because this implies that
parentheses
A bracket is either of two tall fore- or back-facing punctuation marks commonly used to isolate a segment of text or data from its surroundings. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or 'r ...
can be inserted anywhere within such a series of terms, parentheses are usually omitted.
Individual axioms may be "weakened" to assert only the existence of a
left identity and
left inverses. From these ''one-sided axioms'', one can prove that the left identity is also a right identity and a left inverse is also a right inverse for the same element. Since they define exactly the same structures as groups, collectively the axioms are no weaker.
Uniqueness of identity element
The group axioms imply that the identity element is unique: If
and
are identity elements of a group, then
. Therefore, it is customary to speak of ''the'' identity.
Uniqueness of inverses
The group axioms also imply that the inverse of each element is unique: If a group element
has both
and
as inverses, then
Therefore, it is customary to speak of ''the'' inverse of an element.
Division
Given elements
and
of a group
, there is a unique solution
in
to the equation
, namely
. (One usually avoids using fraction notation
unless
is abelian, because of the ambiguity of whether it means
or
.) It follows that for each
in
, the function
that maps each
to
is a
bijection; it is called ''left multiplication by
'' or ''left translation by
''.
Similarly, given
and
, the unique solution to
is
. For each
, the function
that maps each
to
is a bijection called ''right multiplication by
'' or ''right translation by
''.
Basic concepts
When studying sets, one uses concepts such as
subset, function, and
quotient by an equivalence relation
In mathematics, given a category ''C'', a quotient of an object ''X'' by an equivalence relation f: R \to X \times X is a coequalizer for the pair of maps
:R \ \overset\ X \times X \ \overset\ X,\ \ i = 1,2,
where ''R'' is an object in ''C'' and ...
. When studying groups, one uses instead
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 ...
s,
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
s, and
quotient groups. These are the analogues that take the group structure into account.
Group homomorphisms
Group homomorphisms are functions that respect group structure; they may be used to relate two groups. A ''homomorphism'' from a group
to a group
is a function
such that
It would be natural to require also that
respect identities,
, and inverses,
for all
in
. However, these additional requirements need not be included in the definition of homomorphisms, because they are already implied by the requirement of respecting the group operation.
The ''identity homomorphism'' of a group
is the homomorphism
that maps each element of
to itself. An ''inverse homomorphism'' of a homomorphism
is a homomorphism
such that
and
, that is, such that
for all
in
and such that
for all
in
. An ''
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 ...
'' is a homomorphism that has an inverse homomorphism; equivalently, it 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 ...
homomorphism. Groups
and
are called ''isomorphic'' if there exists an isomorphism
. In this case,
can be obtained from
simply by renaming its elements according to the function
; then any statement true for
is true for
, provided that any specific elements mentioned in the statement are also renamed.
The collection of all groups, together with the homomorphisms between them, form 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) ...
, the
category of groups
In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory.
Relation to other categories
There a ...
.
Subgroups
Informally, a ''subgroup'' is a group
contained within a bigger one,
: it has a subset of the elements of
, with the same operation. Concretely, this means that the identity element of
must be contained in
, and whenever
and
are both in
, then so are
and
, so the elements of
, equipped with the group operation on
restricted to
, indeed form a group. In this case, the inclusion map
is a homomorphism.
In the example of symmetries of a square, the identity and the rotations constitute a subgroup
, highlighted in red in the group table of the example: any two rotations composed are still a rotation, and a rotation can be undone by (i.e., is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270°. The
subgroup test provides a
necessary and sufficient condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
for a nonempty subset ''H'' of a group ''G'' to be a subgroup: it is sufficient to check that
for all elements
and
in
. Knowing a group's
subgroups is important in understanding the group as a whole.
Given any subset
of a group
, the subgroup
generated by
consists of all products of elements of
and their inverses. It is the smallest subgroup of
containing
. In the example of symmetries of a square, the subgroup generated by
and
consists of these two elements, the identity element
, and the element
. Again, this is a subgroup, because combining any two of these four elements or their inverses (which are, in this particular case, these same elements) yields an element of this subgroup.
An
injective homomorphism
factors canonically as an isomorphism followed by an inclusion,
for some subgroup of .
Injective homomorphisms are the
monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y.
In the more general setting of category theory, a monomorphism ...
s in the category of groups.
Cosets
In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in the symmetry group of a square, once any reflection is performed, rotations alone cannot return the square to its original position, so one can think of the reflected positions of the square as all being equivalent to each other, and as inequivalent to the unreflected positions; the rotation operations are irrelevant to the question whether a reflection has been performed. Cosets are used to formalize this insight: a subgroup
determines left and right cosets, which can be thought of as translations of
by an arbitrary group element
. In symbolic terms, the ''left'' and ''right'' cosets of
, containing an element
, are
The left cosets of any subgroup
form a
partition
Partition may refer to:
Computing Hardware
* Disk partitioning, the division of a hard disk drive
* Memory partition, a subdivision of a computer's memory, usually for use by a single job
Software
* Partition (database), the division of a ...
of
; that is, the
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
of all left cosets is equal to
and two left cosets are either equal or have an
empty
Empty may refer to:
Music Albums
* ''Empty'' (God Lives Underwater album) or the title song, 1995
* ''Empty'' (Nils Frahm album), 2020
* ''Empty'' (Tait album) or the title song, 2001
Songs
* "Empty" (The Click Five song), 2007
* ...
intersection. The first case
happens
precisely when , i.e., when the two elements differ by an element of
. Similar considerations apply to the right cosets of
. The left cosets of
may or may not be the same as its right cosets. If they are (that is, if all
in
satisfy
), then
is said to be a ''
normal subgroup''.
In
, the group of symmetries of a square, with its subgroup
of rotations, the left cosets
are either equal to
, if
is an element of
itself, or otherwise equal to
(highlighted in green in the group table of
). The subgroup
is normal, because
and similarly for the other elements of the group. (In fact, in the case of
, the cosets generated by reflections are all equal:
.)
Quotient groups
Suppose that
is a normal subgroup of a group
, and
denotes its set of cosets.
Then there is a unique group law on
for which the map
sending each element
to
is a homomorphism.
Explicitly, the product of two cosets
and
is
, the coset
serves as the identity of
, and the inverse of
in the quotient group is .
The group
, read as "
modulo
", is called a ''quotient group'' or ''factor group''.
The quotient group can alternatively be characterized by a
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
.
The elements of the quotient group
are
and
. The group operation on the quotient is shown in the table. For example,
. Both the subgroup
and the quotient
are abelian, but
is not. Sometimes a group can be reconstructed from a subgroup and quotient (plus some additional data), by the
semidirect product construction;
is an example.
The
first isomorphism theorem
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist fo ...
implies that any
surjective homomorphism
factors canonically as a quotient homomorphism followed by an isomorphism:
.
Surjective homomorphisms are the
epimorphism
In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms ,
: g_1 \circ f = g_2 \circ f ...
s in the category of groups.
Presentations
Every group is isomorphic to a quotient of a
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' ...
, in many ways.
For example, the dihedral group
is generated by the right rotation
and the reflection
in a vertical line (every element of
is a finite product of copies of these and their inverses).
Hence there is a surjective homomorphism from the free group
on two generators to
sending
to
and
to
.
Elements in
are called ''relations''; examples include
.
In fact, it turns out that
is the smallest normal subgroup of
containing these three elements; in other words, all relations are consequences of these three.
The quotient of the free group by this normal subgroup is denoted
.
This is called 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 ...
'' of
by generators and relations, because the first isomorphism theorem for yields an isomorphism
.
A presentation of a group can be used to construct 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 ...
, a graphical depiction of a
discrete group
In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and o ...
.
Examples and applications
Examples and applications of groups abound. A starting point is the group
of integers with addition as group operation, introduced above. If instead of addition multiplication is considered, one obtains
multiplicative group
In mathematics and group theory, the term multiplicative group refers to one of the following concepts:
*the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
s. These groups are predecessors of important constructions 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 ...
.
Groups are also applied in many other mathematical areas. Mathematical objects are often examined by
associating groups to them and studying the properties of the corresponding groups. For example,
Henri Poincaré founded what is now called
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 ...
by introducing the
fundamental group. By means of this connection,
topological properties
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological space ...
such as
proximity and
continuity translate into properties of groups. For example, elements of the fundamental group are represented by loops. The second image shows some loops in a plane minus a point. The blue loop is considered
null-homotopic
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
(and thus irrelevant), because it can be
continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point. The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop (or any other loop
winding once around the hole). This way, the fundamental group detects the hole.
In more recent applications, the influence has also been reversed to motivate geometric constructions by a group-theoretical background. In a similar vein,
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 ...
employs geometric concepts, for example in the study of
hyperbolic group
In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a ''word hyperbolic group'' or ''Gromov hyperbolic group'', is a finitely generated group equipped with a word metric satisfying certain properties abstra ...
s. Further branches crucially applying groups include
algebraic geometry and number theory.
In addition to the above theoretical applications, many practical applications of groups exist.
Cryptography
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adver ...
relies on the combination of the abstract group theory approach together with
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 ...
ical knowledge obtained in
computational group theory
In mathematics, computational group theory is the study of
groups by means of computers. It is concerned
with designing and analysing algorithms and
data structures to compute information about groups. The subject
has attracted interest because f ...
, in particular when implemented for finite groups. Applications of group theory are not restricted to mathematics; sciences such as
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
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includi ...
benefit from the concept.
Numbers
Many number systems, such as the integers and the
rationals
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rationa ...
, enjoy a naturally given group structure. In some cases, such as with the rationals, both addition and multiplication operations give rise to group structures. Such number systems are predecessors to more general algebraic structures known 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 ...
and fields. Further abstract algebraic concepts such as
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
s,
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 and
algebras
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
also form groups.
Integers
The group of integers
under addition, denoted
, has been described above. The integers, with the operation of multiplication instead of addition,
do ''not'' form a group. The associativity and identity axioms are satisfied, but inverses do not exist: for example,
is an integer, but the only solution to the equation
in this case is
, which is a rational number, but not an integer. Hence not every element of
has a (multiplicative) inverse.
Rationals
The desire for the existence of multiplicative inverses suggests considering
fractions
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
Fractions of integers (with
nonzero) are known as
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
s. The set of all such irreducible fractions is commonly denoted
. There is still a minor obstacle for
, the rationals with multiplication, being a group: because zero does not have a multiplicative inverse (i.e., there is no
such that
),
is still not a group.
However, the set of all ''nonzero'' rational numbers
does form an abelian group under multiplication, also denoted Associativity and identity element axioms follow from the properties of integers. The closure requirement still holds true after removing zero, because the product of two nonzero rationals is never zero. Finally, the inverse of
is
, therefore the axiom of the inverse element is satisfied.
The rational numbers (including zero) also form a group under addition. Intertwining addition and multiplication operations yields more complicated structures called rings and – if
division
Division or divider may refer to:
Mathematics
*Division (mathematics), the inverse of multiplication
*Division algorithm, a method for computing the result of mathematical division
Military
*Division (military), a formation typically consisting ...
by other than zero is possible, such as in
– fields, which occupy a central position in abstract algebra. Group theoretic arguments therefore underlie parts of the theory of those entities.
Modular arithmetic
![Clock group](https://upload.wikimedia.org/wikipedia/commons/a/a4/Clock_group.svg)
Modular arithmetic for a ''modulus''
defines any two elements
and
that differ by a multiple of
to be equivalent, denoted by
. Every integer is equivalent to one of the integers from
to
, and the operations of modular arithmetic modify normal arithmetic by replacing the result of any operation by its equivalent
representative
Representative may refer to:
Politics
* Representative democracy, type of democracy in which elected officials represent a group of people
* House of Representatives, legislative body in various countries or sub-national entities
* Legislator, som ...
. Modular addition, defined in this way for the integers from
to
, forms a group, denoted as
or
, with
as the identity element and
as the inverse element of
.
A familiar example is addition of hours on the face of a
clock
A clock or a timepiece is a device used to measure and indicate time. The clock is one of the oldest human inventions, meeting the need to measure intervals of time shorter than the natural units such as the day, the lunar month and t ...
, where 12 rather than 0 is chosen as the representative of the identity. If the hour hand is on
and is advanced
hours, it ends up on
, as shown in the illustration. This is expressed by saying that
is congruent to
"modulo
" or, in symbols,
For any prime number
, there is also the
multiplicative group of integers modulo
. Its elements can be represented by
to
. The group operation, multiplication modulo
, replaces the usual product by its representative, the
remainder
In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient ( integer division). In algeb ...
of division by
. For example, for
, the four group elements can be represented by
. In this group,
, because the usual product
is equivalent to
: when divided by
it yields a remainder of
. The primality of
ensures that the usual product of two representatives is not divisible by
, and therefore that the modular product is nonzero. The identity element is represented and associativity follows from the corresponding property of the integers. Finally, the inverse element axiom requires that given an integer
not divisible by
, there exists an integer
such that
that is, such that
evenly divides
. The inverse
can be found by using
Bézout's identity
In mathematics, Bézout's identity (also called Bézout's lemma), named after Étienne Bézout, is the following theorem:
Here the greatest common divisor of and is taken to be . The integers and are called Bézout coefficients for ; they ...
and the fact that the
greatest common divisor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...
In the case
above, the inverse of the element represented by
is that represented by
, and the inverse of the element represented by
is represented , as
. Hence all group axioms are fulfilled. This example is similar to
above: it consists of exactly those elements in the ring
that have a multiplicative inverse. These groups, denoted
, are crucial 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 alg ...
.
Cyclic groups
A ''cyclic group'' is a group all of whose elements are
powers of a particular element
. In multiplicative notation, the elements of the group are
where
means
,
stands for
, etc. Such an element
is called a generator or a
primitive element of the group. In additive notation, the requirement for an element to be primitive is that each element of the group can be written as
In the groups
introduced above, the element
is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are
. Any cyclic group with
elements is isomorphic to this group. A second example for cyclic groups is the group of
th
complex roots of unity, given by
complex number
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 fo ...
s
satisfying
. These numbers can be visualized as the
vertices on a regular
-gon, as shown in blue in the image for
. The group operation is multiplication of complex numbers. In the picture, multiplying with
corresponds to a
counter-clockwise rotation by 60°. From
field theory, the group
is cyclic for prime
: for example, if
,
is a generator since
,
,
, and
.
Some cyclic groups have an infinite number of elements. In these groups, for every non-zero element
, all the powers of
are distinct; despite the name "cyclic group", the powers of the elements do not cycle. An infinite cyclic group is isomorphic to
, the group of integers under addition introduced above. As these two prototypes are both abelian, so are all cyclic groups.
The study of finitely generated abelian groups is quite mature, including the
fundamental theorem of finitely generated abelian groups
In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n ...
; and reflecting this state of affairs, many group-related notions, such as
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
and
commutator, describe the extent to which a given group is not abelian.
Symmetry groups
''Symmetry groups'' are groups consisting of symmetries of given mathematical objects, principally geometric entities, such as the symmetry group of the square given as an introductory example above, although they also arise in algebra such as the symmetries among the roots of polynomial equations dealt with in Galois theory (see below). Conceptually, group theory can be thought of as the study of symmetry.
Symmetries in mathematics greatly simplify the study of
geometrical
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 ...
or
analytical objects. A group is said to
act on another mathematical object ''X'' if every group element can be associated to some operation on ''X'' and the composition of these operations follows the group law. For example, an element of the
(2,3,7) triangle group acts on a triangular
tiling
Tiling may refer to:
*The physical act of laying tiles
* Tessellations
Computing
*The compiler optimization of loop tiling
*Tiled rendering, the process of subdividing an image by regular grid
*Tiling window manager
People
*Heinrich Sylvester T ...
of the
hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P' ...
by permuting the triangles. By a group action, the group pattern is connected to the structure of the object being acted on.
In chemical fields, such as
crystallography,
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 and
point groups describe
molecular symmetries and crystal symmetries. These symmetries underlie the chemical and physical behavior of these systems, and group theory enables simplification of
quantum mechanical
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
analysis of these properties. For example, group theory is used to show that optical transitions between certain quantum levels cannot occur simply because of the symmetry of the states involved.
Group theory helps predict the changes in physical properties that occur when a material undergoes a
phase transition
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states o ...
, for example, from a cubic to a tetrahedral crystalline form. An example is
ferroelectric
Ferroelectricity is a characteristic of certain materials that have a spontaneous electric polarization that can be reversed by the application of an external electric field. All ferroelectrics are also piezoelectric and pyroelectric, with the ad ...
materials, where the change from a paraelectric to a ferroelectric state occurs at the
Curie temperature
In physics and materials science, the Curie temperature (''T''C), or Curie point, is the temperature above which certain materials lose their permanent magnetic properties, which can (in most cases) be replaced by induced magnetism. The Cur ...
and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectric state, accompanied by a so-called soft
phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.
Such
spontaneous symmetry breaking
Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or ...
has found further application in elementary particle physics, where its occurrence is related to the appearance of
Goldstone boson
In particle and condensed matter physics, Goldstone bosons or Nambu–Goldstone bosons (NGBs) are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries. They were discovered by Yoichiro Nambu in par ...
s.
Finite symmetry groups such as the
Mathieu group
In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 obje ...
s are used in
coding theory
Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and data storage. Codes are studied ...
, which is in turn applied in
error correction of transmitted data, and in
CD player
A CD player is an electronic device that plays audio compact discs, which are a digital optical disc data storage format. CD players were first sold to consumers in 1982. CDs typically contain recordings of audio material such as music or audio ...
s. Another application is
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 ...
, which characterizes functions having
antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolicall ...
s of a prescribed form, giving group-theoretic criteria for when solutions of certain
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s are well-behaved. Geometric properties that remain stable under group actions are investigated in
(geometric) invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descri ...
.
General linear group and representation theory
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 consist of
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 ...
together with
matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
. The ''general linear group''
consists of all
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
-by-
matrices with real entries. Its subgroups are referred to as ''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''. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the
special orthogonal group . It describes all possible rotations in
dimensions.
Rotation matrices in this group are used in
computer graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
.
''Representation theory'' is both an application of the group concept and important for a deeper understanding of groups. It studies the group by its group actions on other spaces. A broad class of
group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to ...
s are linear representations in which the group acts on a vector space, such as the three-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 ...
. A representation of a group
on an
-
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
al real vector space is simply a group homomorphism
from the group to the general linear group. This way, the group operation, which may be abstractly given, translates to the multiplication of matrices making it accessible to explicit computations.
A group action gives further means to study the object being acted on. On the other hand, it also yields information about the group. Group representations are an organizing principle in the theory of finite groups, Lie groups, algebraic groups and
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 ...
s, especially (locally)
compact group
In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...
s.
Galois groups
''Galois groups'' were developed to help solve polynomial equations by capturing their symmetry features. For example, the solutions of the
quadratic equation
In algebra, a quadratic equation () is any equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not q ...
are given by
Each solution can be obtained by replacing the
sign by
or
; analogous formulae are known for
cubic and
quartic equation
In mathematics, a quartic equation is one which can be expressed as a ''quartic function'' equaling zero. The general form of a quartic equation is
:ax^4+bx^3+cx^2+dx+e=0 \,
where ''a'' ≠ 0.
The quartic is the highest order polynomi ...
s, but do ''not'' exist in general for
degree 5 and higher. In the quadratic formula, changing the sign (permuting the resulting two solutions) can be viewed as a (very simple) group operation. Analogous Galois groups act on the solutions of higher-degree polynomials and are closely related to the existence of formulas for their solution. Abstract properties of these groups (in particular their
solvability) give a criterion for the ability to express the solutions of these polynomials using solely addition, multiplication, and
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
similar to the formula above.
Modern
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 ...
generalizes the above type of Galois groups by shifting to field theory and considering
field extensions formed as the
splitting field
In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors.
Definition
A splitting field of a poly ...
of a polynomial. This theory establishes—via the
fundamental theorem of Galois theory
In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in his development of Galois theory.
In its most basi ...
—a precise relationship between fields and groups, underlining once again the ubiquity of groups in mathematics.
Finite groups
A group is called ''finite'' if it has a
finite number of elements. The number of elements is called the
order of the group. An important class 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''
, the groups of permutations of
objects. For example, the dihedral group of order 6, symmetric group on 3 letters
is the group of all possible reorderings of the objects. The three letters ABC can be reordered into ABC, ACB, BAC, BCA, CAB, CBA, forming in total 6 (factorial of 3) elements. The group operation is composition of these reorderings, and the identity element is the reordering operation that leaves the order unchanged. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group
for a suitable integer
, according to Cayley's theorem. Parallel to the group of symmetries of the square above,
can also be interpreted as the group of symmetries of an equilateral triangle.
The order of an element
in a group
is the least positive integer
such that
, where
represents
that is, application of the operation "
" to
copies of
. (If "
" represents multiplication, then
corresponds to the
th power of
.) In infinite groups, such an
may not exist, in which case the order of
is said to be infinity. The order of an element equals the order of the cyclic subgroup generated by this element.
More sophisticated counting techniques, for example, counting cosets, yield more precise statements about finite groups: Lagrange's theorem (group theory), Lagrange's Theorem states that for a finite group
the order of any finite subgroup
divisor, divides the order of
. The Sylow theorems give a partial converse.
The dihedral group
of symmetries of a square is a finite group of order 8. In this group, the order of
is 4, as is the order of the subgroup
that this element generates. The order of the reflection elements
etc. is 2. Both orders divide 8, as predicted by Lagrange's theorem. The groups
of multiplication modulo a prime
have order
.
Finite abelian groups
Any finite abelian group is isomorphic to a direct product, product of finite cyclic groups; this statement is part of the
fundamental theorem of finitely generated abelian groups
In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n ...
.
Any group of prime order
is isomorphic to the cyclic group
(a consequence of Lagrange's theorem_(group theory), Lagrange's theorem).
Any group of order
is abelian, isomorphic to
or
.
But there exist nonabelian groups of order
; the dihedral group
of order
above is an example.
Simple groups
When a group
has a normal subgroup
other than
and
itself, questions about
can sometimes be reduced to questions about
and
. A nontrivial group is called ''simple group, simple'' if it has no such normal subgroup. Finite simple groups are to finite groups as prime numbers are to positive integers: they serve as building blocks, in a sense made precise by the Jordan–Hölder theorem.
Classification of finite simple groups
Computer algebra systems have been used to List of small groups, list all groups of order up to 2000.
But classification theorems, classifying all finite groups is a problem considered too hard to be solved.
The classification of all finite ''simple'' groups was a major achievement in contemporary group theory. There are List of finite simple groups, several infinite families of such groups, as well as 26 "sporadic groups" that do not belong to any of the families. The largest sporadic group is called the monster group. The monstrous moonshine conjectures, proved by Richard Borcherds, relate the monster group to certain modular functions.
The gap between the classification of simple groups and the classification of all groups lies in the extension problem.
[.]
Groups with additional structure
An equivalent definition of group consists of replacing the "there exist" part of the group axioms by operations whose result is the element that must exist. So, a group is a set
equipped with a binary operation
(the group operation), a unary operation
(which provides the inverse) and a nullary operation, which has no operand and results in the identity element. Otherwise, the group axioms are exactly the same. This variant of the definition avoids existential quantifiers and is used in computing with groups and for computer-aided proofs.
This way of defining groups lends itself to generalizations such as the notion of group object in a category. Briefly, this is an object with morphisms that mimic the group axioms.
Topological groups
Some topological spaces may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions; informally,
and
must not vary wildly if
and
vary only a little. Such groups are called ''topological groups,'' and they are the group objects in the category of topological spaces. The most basic examples are the group of real numbers under addition and the group of nonzero real numbers under multiplication. Similar examples can be formed from any other topological field, such as the field of complex numbers or the field of p-adic number, -adic numbers. These examples are locally compact topological group, locally compact, so they have Haar measures and can be studied via harmonic analysis. Other locally compact topological groups include the group of points of an algebraic group over a local field or adele ring; these are basic to number theory Galois groups of infinite algebraic field extensions are equipped with the Krull topology, which plays a role in Fundamental theorem of Galois theory#Infinite case, infinite Galois theory. A generalization used in algebraic geometry is the étale fundamental group.
Lie groups
A ''Lie group'' is a group that also has the structure of a differentiable manifold; informally, this means that it diffeomorphism, looks locally like a Euclidean space of some fixed dimension. Again, the definition requires the additional structure, here the manifold structure, to be compatible: the multiplication and inverse maps are required to be smooth map, smooth.
A standard example is the general linear group introduced above: it is an open subset of the space of all
-by-
matrices, because it is given by the inequality
where
denotes an
-by-
matrix.
Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time, are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description. Another example is the group of Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation—as a model of
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
in
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws ...
. The full symmetry group of Minkowski space, i.e., including translations, is known as 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 ...
. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. Local symmetry, Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory. An important example of a gauge theory is the
Standard Model, which describes three of the four known fundamental forces and classifies all known elementary particles.
Generalizations
More general structures may be defined by relaxing some of the axioms defining a group. The table gives a list of several structures generalizing groups.
For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers
(including zero) under addition form a monoid, as do the nonzero integers under multiplication
. Adjoining inverses of all elements of the monoid
produces a group
, and likewise adjoining inverses to any (abelian) monoid produces a group known as the Grothendieck group of .
A group can be thought of as a small category with one object in which every morphism is an isomorphism: given such a category, the set
is a group; conversely, given a group , one can build a small category with one object in which
.
More generally, a groupoid is any small category in which every morphism is an isomorphism.
In a groupoid, the set of all morphisms in the category is usually not a group, because the composition is only partially defined: is defined only when the source of matches the target of .
Groupoids arise in topology (for instance, the fundamental groupoid) and in the theory of stack (mathematics), stacks.
Finally, it is possible to generalize any of these concepts by replacing the binary operation with an arity, -ary operation (i.e., an operation taking arguments, for some nonnegative integer ). With the proper generalization of the group axioms, this gives a notion of n-ary group, -ary group.
See also
* List of group theory topics
Notes
Citations
References
General references
* , Chapter 2 contains an undergraduate-level exposition of the notions covered in this article.
*
* , an elementary introduction.
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Special references
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Historical references
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* (Galois work was first published by Joseph Liouville in 1843).
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External links
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{{DEFAULTSORT:Group (Mathematics)
Group theory, *
Algebraic structures
Symmetry