In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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-Man ...
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
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when a ...
. These three
axioms hold for
number systems and many other mathematical structures. For example, the
integers 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 roots. 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 groups arise naturally in the study of
symmetries and
geometric transformations: 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. The
Poincaré group is a Lie group consisting of the symmetries of
spacetime in
special relativity.
Point groups describe
symmetry in molecular chemistry.
The concept of a group arose in the study of
polynomial equations, starting with
Évariste Galois 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:
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* ''The Root'' (magazine), an online magazine focusing ...
of an equation, now called a
Galois group. After contributions from other fields such as
number theory and geometry, the group notion was generalized and firmly established around 1870. Modern
group theory—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
subgroups,
quotient groups and
simple groups. 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 (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, completed in 2004. Since the mid-1980s,
geometric group theory, which studies
finitely generated groups 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
integers
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''.
*If
is any integer, then
and
.
Zero is called the ''
identity element'' 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'' 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. Th ...
. 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 objects. 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 numbers
, which has the operations of addition
and
multiplication . Formally,
is a set,
is a group, and
is a
field. 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, and the group is called an
abelian group. 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 groups,
symmetry groups,
permutation groups, and
automorphism groups, 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 are
congruent if one can be changed into the other using a combination of
rotations,
reflections, and
translations. 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. A
square has eight symmetries. These are:
* the
identity operation
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
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
diagonals (
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 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 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
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of
polynomial equations of degree higher than 4. The 19th-century French mathematician
Évariste Galois, extending prior work of
Paolo Ruffini 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 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.
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's 1872
Erlangen program. After novel geometries such as
hyperbolic and
projective geometry 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 S ...
founded the study of
Lie groups in 1884.
The third field contributing to group theory was
number theory. Certain abelian group structures had been used implicitly in
Carl Friedrich Gauss's number-theoretical work ''
Disquisitiones Arithmeticae'' (1798), and more explicitly by
Leopold Kronecker. In 1847,
Ernst Kummer made early attempts to prove
Fermat's Last Theorem 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 (mathematics), 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 ...
s.
The convergence of these various sources into a uniform theory of groups started with
Camille Jordan's (1870).
Walther von Dyck (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 res ...
, who worked on
representation theory of finite groups,
Richard Brauer'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 ha ...
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 ...
's papers. The theory of Lie groups, and more generally
locally compact groups was studied by
Hermann Weyl,
Élie Cartan and many others. Its
algebraic counterpart, the theory of
algebraic groups, 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 fou ...
(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 ...
and
Jacques Tits.
The
University of Chicago'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 dissert ...
,
John G. Thompson and
Walter Feit
Walter Feit (October 26, 1930 – July 29, 2004) was an Austrian-born American mathematician who worked in finite group theory and representation theory. His contributions provided elementary infrastructure used in algebra, geometry, topolo ...
, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the
classification of finite simple groups, 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 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
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 s ...
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
subgroups,
homomorphisms, 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'' is a homomorphism that has an inverse homomorphism; equivalently, it is a
bijective 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, the
category of groups.
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
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 subgrou ...
provides a
necessary and sufficient condition 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
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 subgrou ...
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
monomorphisms 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 of
; that is, the
union 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.
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 implies that any
surjective homomorphism
factors canonically as a quotient homomorphism followed by an isomorphism:
.
Surjective homomorphisms are the
epimorphisms in the category of groups.
Presentations
Every group is isomorphic to a quotient of a
free group, 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'' 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, a graphical depiction of a
discrete group.
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 groups. 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 groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
.
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 by introducing the
fundamental group. By means of this connection,
topological properties 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 (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 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 relies on the combination of the abstract group theory approach together with
algorithmical 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,
chemistry
Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, proper ...
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 Applied science, practical discipli ...
benefit from the concept.
Numbers
Many number systems, such as the integers and the
rationals, 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 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
* Modul ...
s,
vector spaces and
algebras 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
Fractions of integers (with
nonzero) are known as
rational numbers. 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 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
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. 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, 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 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 and the fact that the
greatest common divisor 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.
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 numbers
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, ...
; and reflecting this state of affairs, many group-related notions, such as
center 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 c ...
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 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 ' ...
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 groups 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 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, for example, from a cubic to a tetrahedral crystalline form. An example is
ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the
Curie temperature 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 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 part ...
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, which is in turn applied in
error correction of transmitted data, and in
CD players. Another application is
differential Galois theory, which characterizes functions having
antiderivatives of a prescribed form, giving group-theoretic criteria for when solutions of certain
differential equations are well-behaved. Geometric properties that remain stable under group actions are investigated in
(geometric) invariant theory.
General linear group and representation theory
Matrix groups consist of
matrices together with
matrix multiplication. The ''general linear group''
consists of all
invertible -by-
matrices with real entries. Its subgroups are referred to as ''matrix groups'' or ''
linear groups''. 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 representations are linear representations in which the group acts on a vector space, such as the three-dimensional
Euclidean space . A representation of a group
on an
-
dimensional 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 groups, especially (locally)
compact groups.
Galois groups
''Galois groups'' were developed to help solve polynomial equations by capturing their symmetry features. For example, the solutions of the
quadratic equation are given by
Each solution can be obtained by replacing the
sign by
or
; analogous formulae are known for
cubic and
quartic equations, 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 generalizes the above type of Galois groups by shifting to field theory and considering
field extensions formed as the
splitting field 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
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
of the group. An important class is the ''
symmetric groups''
, the groups of permutations of
objects. For example, the
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 states that for a finite group
the order of any finite subgroup
divides
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible b ...
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
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, ...
.
Any group of prime order
is isomorphic to the cyclic group
(a consequence of
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
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 ...
'' 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 In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many natura ...
.
Classification of finite simple groups
Computer algebra systems have been used to
list all groups of order up to 2000.
But
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
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
Richard Ewen Borcherds (; born 29 November 1959) is a British mathematician currently working in quantum field theory. He is known for his work in lattices, group theory, and infinite-dimensional algebras, for which he was awarded the Fields Me ...
, 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 proof
A computer-assisted proof is a mathematical proof that has been at least partially generated by computer.
Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of a mathematical theorem. The idea is to use ...
s.
This way of defining groups lends itself to generalizations such as the notion of
group object In category theory, a branch of mathematics, group objects are certain generalizations of groups that are built on more complicated structures than sets. A typical example of a group object is a topological group, a group whose underlying set is ...
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
-adic numbers. These examples are
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
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
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.
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
special relativity. The full symmetry group of Minkowski space, i.e., including translations, is known as the
Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for
quantum field theories
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles ...
.
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 In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. It can be viewed as an extension of the more widely-known fundamental group; as such, it captures information about the homotopy type of a ...
) and in the theory of
stacks.
Finally, it is possible to generalize any of these concepts by replacing the binary operation with an
-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
-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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*
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*
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*
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*
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* .
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*
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*
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*
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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
*
{{DEFAULTSORT:Group (Mathematics)
*
Algebraic structures
Symmetry