TheInfoList

OR:

In
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
, group theory studies the
algebraic structure In mathematics, an algebraic structure consists of a nonempty Set (mathematics), set ''A'' (called the underlying set, carrier set or domain), a collection of operation (mathematics), operations on ''A'' (typically binary operations such as addit ...
s known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and
vector space 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 ...
s, can all be seen as groups endowed with additional operations and
axiom An axiom, postulate, or assumption is a statement (logic), statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that whi ...
s. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macrosc ...
s and the
hydrogen atom A hydrogen atom is an atom of the chemical element hydrogen. The Electric charge, electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic ...
, and three of the four known fundamental forces in the universe, may be modelled by
symmetry group In group theory, the symmetry group of a geometric object is the group (mathematics), group of all Transformation (geometry), transformations under which the object is invariant (mathematics), invariant, endowed with the group operation of Fu ...
s. Thus group theory and the closely related representation theory have many important applications in
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
,
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, properties ...
, and
materials science Materials science is an interdisciplinary Interdisciplinarity or interdisciplinary studies involves the combination of multiple academic disciplines into one activity (e.g., a research project). It draws knowledge from several other field ...
. Group theory is also central to public key cryptography. The early history of group theory dates from the 19th century. One of the most important mathematical achievements of the 20th century was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 2004, that culminated in a complete
classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory stating that every List of finite simple groups, finite simple group is either cyclic groups, cyclic, or alternating groups, alternating, or it belongs to ...
.

# History

Group theory has three main historical sources:
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative intege ...
, the theory of algebraic equations, and
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
. The number-theoretic strand was begun by
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields. Early results about permutation groups were obtained by Lagrange, Ruffini, and Abel in their quest for general solutions of polynomial equations of high degree. Évariste Galois coined the term "group" and established a connection, now known as Galois theory, between the nascent theory of groups and field theory. In geometry, groups first became important in projective geometry and, later, non-Euclidean geometry. Felix Klein's Erlangen program proclaimed group theory to be the organizing principle of geometry. Galois, in the 1830s, was the first to employ groups to determine the solvability of
polynomial equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For many authors, the term '' ...
s. Arthur Cayley and Augustin Louis Cauchy pushed these investigations further by creating the theory of permutation groups. The second historical source for groups stems from geometrical situations. In an attempt to come to grips with possible geometries (such as euclidean, hyperbolic or projective geometry) using group theory, Felix Klein initiated the Erlangen programme. Sophus Lie, in 1884, started using groups (now called Lie groups) attached to analytic problems. Thirdly, groups were, at first implicitly and later explicitly, used in
algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
. The different scope of these early sources resulted in different notions of groups. The theory of groups was unified starting around 1880. Since then, the impact of group theory has been ever growing, giving rise to the birth of
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
in the early 20th century, representation theory, and many more influential spin-off domains. The
classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory stating that every List of finite simple groups, finite simple group is either cyclic groups, cyclic, or alternating groups, alternating, or it belongs to ...
is a vast body of work from the mid 20th century, classifying all the finite
simple group SIMPLE Group Limited is a list of conglomerates, 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 Cor ...
s.

# Main classes of groups

The range of groups being considered has gradually expanded from finite permutation groups and special examples of matrix groups to abstract groups that may be specified through a
presentation A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presen ...
by generators and relations.

## Permutation groups

The first class of groups to undergo a systematic study was permutation groups. Given any set ''X'' and a collection ''G'' of
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is pair ...
s of ''X'' into itself (known as ''permutations'') that is closed under compositions and inverses, ''G'' is a group
acting Acting is an activity in which a story is told by means of its Enactment (psychology), enactment by an actor or actress who adopts a Character (arts), character—in theatre, television, film, radio, or any other medium that makes use of the ...
on ''X''. If ''X'' consists of ''n'' elements and ''G'' consists of ''all'' permutations, ''G'' is the symmetric group S''n''; in general, any permutation group ''G'' is a
subgroup In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ...
of the symmetric group of ''X''. An early construction due to Cayley exhibited any group as a permutation group, acting on itself () by means of the left regular representation. In many cases, the structure of a permutation group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for , the alternating group A''n'' is simple, i.e. does not admit any proper normal subgroups. This fact plays a key role in the impossibility of solving a general algebraic equation of degree in radicals.

## Matrix groups

The next important class of groups is given by ''matrix groups'', or linear groups. Here ''G'' is a set consisting of invertible matrices of given order ''n'' over a field ''K'' that is closed under the products and inverses. Such a group acts on the ''n''-dimensional vector space ''K''''n'' by
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
s. This action makes matrix groups conceptually similar to permutation groups, and the geometry of the action may be usefully exploited to establish properties of the group ''G''.

## Transformation groups

Permutation groups and matrix groups are special cases of
transformation group In mathematics, the automorphism group of an object ''X'' is the group (mathematics), group consisting of automorphisms of ''X'' under Function composition, composition of morphisms. For example, if ''X'' is a Dimension (vector space), finite-dime ...
s: groups that act on a certain space ''X'' preserving its inherent structure. In the case of permutation groups, ''X'' is a set; for matrix groups, ''X'' is a
vector space In mathematics Mathematics 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 ...
. The concept of a transformation group is closely related with the concept of a
symmetry group In group theory, the symmetry group of a geometric object is the group (mathematics), group of all Transformation (geometry), transformations under which the object is invariant (mathematics), invariant, endowed with the group operation of Fu ...
: transformation groups frequently consist of ''all'' transformations that preserve a certain structure. The theory of transformation groups forms a bridge connecting group theory with
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra ...
. A long line of research, originating with Lie and Klein, considers group actions on manifolds by homeomorphisms or diffeomorphisms. The groups themselves may be discrete or continuous.

## Abstract groups

Most groups considered in the first stage of the development of group theory were "concrete", having been realized through numbers, permutations, or matrices. It was not until the late nineteenth century that the idea of an abstract group as a set with operations satisfying a certain system of axioms began to take hold. A typical way of specifying an abstract group is through a
presentation 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. Presen ...
by ''generators and relations'', : $G = \langle S, R\rangle.$ A significant source of abstract groups is given by the construction of a ''factor group'', or quotient group, ''G''/''H'', of a group ''G'' by a normal subgroup ''H''. Class groups of algebraic number fields were among the earliest examples of factor groups, of much interest in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative intege ...
. If a group ''G'' is a permutation group on a set ''X'', the factor group ''G''/''H'' is no longer acting on ''X''; but the idea of an abstract group permits one not to worry about this discrepancy. The change of perspective from concrete to abstract groups makes it natural to consider properties of groups that are independent of a particular realization, or in modern language, invariant under
isomorphism In mathematics, an isomorphism is a structure-preserving Map (mathematics), mapping between two Mathematical structure, structures of the same type that can be reversed by an inverse function, inverse mapping. Two mathematical structures are iso ...
, as well as the classes of group with a given such property:
finite group Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
s, periodic groups,
simple group SIMPLE Group Limited is a list of conglomerates, 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 Cor ...
s, solvable groups, and so on. Rather than exploring properties of an individual group, one seeks to establish results that apply to a whole class of groups. The new paradigm was of paramount importance for the development of mathematics: it foreshadowed the creation of
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
in the works of Hilbert,
Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrians, Austrian mathematician of Armenians, Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number th ...
, Emmy Noether, and mathematicians of their school.

An important elaboration of the concept of a group occurs if ''G'' is endowed with additional structure, notably, of a topological space, differentiable manifold, or algebraic variety. If the group operations ''m'' (multiplication) and ''i'' (inversion), : $m: G\times G\to G, \left(g,h\right)\mapsto gh, \quad i:G\to G, g\mapsto g^,$ are compatible with this structure, that is, they are continuous, smooth or regular (in the sense of algebraic geometry) maps, then ''G'' is a topological group, a Lie group, or an
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group (mathematics), group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry a ...
. The presence of extra structure relates these types of groups with other mathematical disciplines and means that more tools are available in their study. Topological groups form a natural domain for abstract harmonic analysis, whereas Lie groups (frequently realized as transformation groups) are the mainstays of
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra ...
and unitary representation theory. Certain classification questions that cannot be solved in general can be approached and resolved for special subclasses of groups. Thus, compact connected Lie groups have been completely classified. There is a fruitful relation between infinite abstract groups and topological groups: whenever a group ''Γ'' can be realized as a lattice in a topological group ''G'', the geometry and analysis pertaining to ''G'' yield important results about ''Γ''. A comparatively recent trend in the theory of finite groups exploits their connections with compact topological groups ( profinite groups): for example, a single ''p''-adic analytic group ''G'' has a family of quotients which are finite ''p''-groups of various orders, and properties of ''G'' translate into the properties of its finite quotients.

# Branches of group theory

## Finite group theory

During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups. As a consequence, the complete
classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory stating that every List of finite simple groups, finite simple group is either cyclic groups, cyclic, or alternating groups, alternating, or it belongs to ...
was achieved, meaning that all those
simple group SIMPLE Group Limited is a list of conglomerates, 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 Cor ...
s from which all finite groups can be built are now known. During the second half of the twentieth century, mathematicians such as Chevalley and Steinberg also increased our understanding of finite analogs of
classical group 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 m ...
s, and other related groups. One such family of groups is the family of
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrix, invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group (mathematics), group, because the product of two in ...
s over finite fields. Finite groups often occur when considering
symmetry 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 ...
of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. The theory of Lie groups, which may be viewed as dealing with " continuous symmetry", is strongly influenced by the associated Weyl groups. These are finite groups generated by reflections which act on a finite-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
. The properties of finite groups can thus play a role in subjects such as
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict List of natural phenomena, natural phenomena. This is in contrast to experimental ph ...
and
chemistry Chemistry is the 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, properties ...
.

## Representation of groups

Saying that a group ''G'' '' acts'' on a set ''X'' means that every element of ''G'' defines a bijective map on the set ''X'' in a way compatible with the group structure. When ''X'' has more structure, it is useful to restrict this notion further: a representation of ''G'' on a
vector space In mathematics Mathematics 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 ...
''V'' is a
group homomorphism In mathematics, given two group (mathematics), groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function (mathematics), function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' ...
: :$\rho:G \to \operatorname\left(V\right),$ where GL(''V'') consists of the invertible linear transformations of ''V''. In other words, to every group element ''g'' is assigned an
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of map (mathematics), mapping the object to itself while preserving all of its structure. The Set (m ...
''ρ''(''g'') such that for any ''h'' in ''G''. This definition can be understood in two directions, both of which give rise to whole new domains of mathematics. On the one hand, it may yield new information about the group ''G'': often, the group operation in ''G'' is abstractly given, but via ''ρ'', it corresponds to the multiplication of matrices, which is very explicit. On the other hand, given a well-understood group acting on a complicated object, this simplifies the study of the object in question. For example, if ''G'' is finite, it is known that ''V'' above decomposes into irreducible parts (see Maschke's theorem). These parts, in turn, are much more easily manageable than the whole ''V'' (via Schur's lemma). Given a group ''G'', representation theory then asks what representations of ''G'' exist. There are several settings, and the employed methods and obtained results are rather different in every case: representation theory of finite groups and representations of Lie groups are two main subdomains of the theory. The totality of representations is governed by the group's characters. For example, Fourier polynomials can be interpreted as the characters of U(1), the group of complex numbers of
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign (mathematics), sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative number, negative (in which cas ...
''1'', acting on the ''L''2-space of periodic functions.

## Lie theory

A Lie group is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie, who laid the foundations of the theory of continuous
transformation group In mathematics, the automorphism group of an object ''X'' is the group (mathematics), group consisting of automorphisms of ''X'' under Function composition, composition of morphisms. For example, if ''X'' is a Dimension (vector space), finite-dime ...
s. The term ''groupes de Lie'' first appeared in French in 1893 in the thesis of Lie's student Arthur Tresse, page 3. Lie groups represent the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict List of natural phenomena, natural phenomena. This is in contrast to experimental ph ...
. They provide a natural framework for analysing the continuous symmetries of differential equations ( differential Galois theory), in much the same way as permutation groups are used in Galois theory for analysing the discrete symmetries of algebraic equations. An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations.

## Combinatorial and geometric group theory

Groups can be described in different ways. Finite groups can be described by writing down the group table consisting of all possible multiplications . A more compact way of defining a group is by ''generators and relations'', also called the ''presentation'' of a group. Given any set ''F'' of generators $\_$, the free group generated by ''F'' surjects onto the group ''G''. The kernel of this map is called the subgroup of relations, generated by some subset ''D''. The presentation is usually denoted by $\langle F \mid D\rangle.$ For example, the group presentation $\langle a,b\mid aba^b^\rangle$ describes a group which is isomorphic to $\mathbb\times\mathbb.$ A string consisting of generator symbols and their inverses is called a ''word''. Combinatorial group theory studies groups from the perspective of generators and relations. It is particularly useful where finiteness assumptions are satisfied, for example finitely generated groups, or finitely presented groups (i.e. in addition the relations are finite). The area makes use of the connection of graphs via their
fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It recor ...
s. For example, one can show that every subgroup of a free group is free. There are several natural questions arising from giving a group by its presentation. The ''word problem for groups, word problem'' asks whether two words are effectively the same group element. By relating the problem to Turing machines, one can show that there is in general no algorithm solving this task. Another, generally harder, algorithmically insoluble problem is the group isomorphism problem, which asks whether two groups given by different presentations are actually isomorphic. For example, the group with presentation $\langle x,y \mid xyxyx = e \rangle,$ is isomorphic to the additive group Z of integers, although this may not be immediately apparent. (Writing $z=xy$, one has $G \cong \langle z,y \mid z^3 = y\rangle \cong \langle z\rangle.$) Geometric group theory attacks these problems from a geometric viewpoint, either by viewing groups as geometric objects, or by finding suitable geometric objects a group acts on. The first idea is made precise by means of the Cayley graph, whose vertices correspond to group elements and edges correspond to right multiplication in the group. Given two elements, one constructs the word metric given by the length of the minimal path between the elements. A theorem of John Milnor, Milnor and Svarc then says that given a group ''G'' acting in a reasonable manner on a metric space ''X'', for example a compact manifold, then ''G'' is quasi-isometry, quasi-isometric (i.e. looks similar from a distance) to the space ''X''.

# Connection of groups and symmetry

Given a structured object ''X'' of any sort, a
symmetry Symmetry (from 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 ...
is a mapping of the object onto itself which preserves the structure. This occurs in many cases, for example *If ''X'' is a set with no additional structure, a symmetry is a bijection, bijective map from the set to itself, giving rise to permutation groups. *If the object ''X'' is a set of points in the plane with its metric (mathematics), metric structure or any other metric space, a symmetry is a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is pair ...
of the set to itself which preserves the distance between each pair of points (an isometry). The corresponding group is called isometry group of ''X''. *If instead angles are preserved, one speaks of conformal maps. Conformal maps give rise to Kleinian groups, for example. *Symmetries are not restricted to geometrical objects, but include algebraic objects as well. For instance, the equation $x^2-3=0$ has the two solutions $\sqrt$ and $-\sqrt$. In this case, the group that exchanges the two roots is the Galois group belonging to the equation. Every polynomial equation in one variable has a Galois group, that is a certain permutation group on its roots. The axioms of a group formalize the essential aspects of
symmetry Symmetry (from 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 ...
. Symmetries form a group: they are closure (mathematics), closed because if you take a symmetry of an object, and then apply another symmetry, the result will still be a symmetry. The identity keeping the object fixed is always a symmetry of an object. Existence of inverses is guaranteed by undoing the symmetry and the associativity comes from the fact that symmetries are functions on a space, and composition of functions is associative. Frucht's theorem says that every group is the symmetry group of some Graph (discrete mathematics), graph. So every abstract group is actually the symmetries of some explicit object. The saying of "preserving the structure" of an object can be made precise by working in a category (mathematics), category. Maps preserving the structure are then the morphisms, and the symmetry group is the automorphism group of the object in question.

# Applications of group theory

Applications of group theory abound. Almost all structures in
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
are special cases of groups. Ring (mathematics), Rings, for example, can be viewed as abelian groups (corresponding to addition) together with a second operation (corresponding to multiplication). Therefore, group theoretic arguments underlie large parts of the theory of those entities.

## Galois theory

Galois theory uses groups to describe the symmetries of the roots of a polynomial (or more precisely the automorphisms of the algebras generated by these roots). The fundamental theorem of Galois theory provides a link between algebraic field extensions and group theory. It gives an effective criterion for the solvability of polynomial equations in terms of the solvability of the corresponding Galois group. For example, ''S''5, the symmetric group in 5 elements, is not solvable which implies that the general quintic equation cannot be solved by radicals in the way equations of lower degree can. The theory, being one of the historical roots of group theory, is still fruitfully applied to yield new results in areas such as class field theory.

## Algebraic topology

Algebraic topology is another domain which prominently functor, associates groups to the objects the theory is interested in. There, groups are used to describe certain invariants of topological spaces. They are called "invariants" because they are defined in such a way that they do not change if the space is subjected to some homeomorphism, deformation. For example, the
fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It recor ...
"counts" how many paths in the space are essentially different. The Poincaré conjecture, proved in 2002/2003 by Grigori Perelman, is a prominent application of this idea. The influence is not unidirectional, though. For example, algebraic topology makes use of Eilenberg–MacLane spaces which are spaces with prescribed homotopy groups. Similarly algebraic K-theory relies in a way on classifying spaces of groups. Finally, the name of the torsion subgroup of an infinite group shows the legacy of topology in group theory.

## Algebraic geometry

Algebraic geometry likewise uses group theory in many ways. Abelian variety, Abelian varieties have been introduced above. The presence of the group operation yields additional information which makes these varieties particularly accessible. They also often serve as a test for new conjectures. (For example the Hodge conjecture (in certain cases).) The one-dimensional case, namely elliptic curves is studied in particular detail. They are both theoretically and practically intriguing. In another direction, toric variety, toric varieties are algebraic variety, algebraic varieties acted on by a torus. Toroidal embeddings have recently led to advances in algebraic geometry, in particular resolution of singularities.

## Algebraic number theory

Algebraic number theory makes uses of groups for some important applications. For example, Euler product, Euler's product formula, :$\begin \sum_\frac& = \prod_ \frac, \\ \end \!$ captures Fundamental theorem of arithmetic, the fact that any integer decomposes in a unique way into prime number, primes. The failure of this statement for Dedekind ring, more general rings gives rise to class groups and regular primes, which feature in Ernst Kummer, Kummer's treatment of Fermat's Last Theorem.

## Harmonic analysis

Analysis on Lie groups and certain other groups is called harmonic analysis. Haar measures, that is, integrals invariant under the translation in a Lie group, are used for pattern recognition and other image processing techniques.

## Combinatorics

In combinatorics, the notion of permutation group and the concept of group action are often used to simplify the counting of a set of objects; see in particular Burnside's lemma.

## Music

The presence of the 12-Periodic group, periodicity in the circle of fifths yields applications of elementary group theory in set theory (music), musical set theory. Transformational theory models musical transformations as elements of a mathematical group.

## Physics

In
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
, groups are important because they describe the symmetries which the laws of physics seem to obey. According to Noether's theorem, every continuous symmetry of a physical system corresponds to a Conservation law (physics), conservation law of the system. Physicists are very interested in group representations, especially of Lie groups, since these representations often point the way to the "possible" physical theories. Examples of the use of groups in physics include the Standard Model, gauge theory, the Lorentz group, and the Poincaré group. Group theory can be used to resolve the incompleteness of the statistical interpretations of mechanics developed by Josiah Willard Gibbs, Willard Gibbs, relating to the summing of an infinite number of probabilities to yield a meaningful solution.Norbert Wiener, Cybernetics: Or Control and Communication in the Animal and the Machine, , Ch 2

## Chemistry and materials science

In
chemistry Chemistry is the 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, properties ...
and
materials science Materials science is an interdisciplinary Interdisciplinarity or interdisciplinary studies involves the combination of multiple academic disciplines into one activity (e.g., a research project). It draws knowledge from several other field ...
, point groups are used to classify regular polyhedra, and the molecular symmetry, symmetries of molecules, and space groups to classify crystal structures. The assigned groups can then be used to determine physical properties (such as chemical polarity and Chirality (chemistry), chirality), spectroscopic properties (particularly useful for Raman spectroscopy, infrared spectroscopy, circular dichroism spectroscopy, magnetic circular dichroism spectroscopy, UV/Vis spectroscopy, and fluorescence spectroscopy), and to construct molecular orbitals. Molecular symmetry is responsible for many physical and spectroscopic properties of compounds and provides relevant information about how chemical reactions occur. In order to assign a point group for any given molecule, it is necessary to find the set of symmetry operations present on it. The symmetry operation is an action, such as a rotation around an axis or a reflection through a mirror plane. In other words, it is an operation that moves the molecule such that it is indistinguishable from the original configuration. In group theory, the rotation axes and mirror planes are called "symmetry elements". These elements can be a point, line or plane with respect to which the symmetry operation is carried out. The symmetry operations of a molecule determine the specific point group for this molecule. In
chemistry Chemistry is the 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, properties ...
, there are five important symmetry operations. They are identity operation (E), rotation operation or proper rotation (C''n''), reflection operation (σ), inversion (i) and rotation reflection operation or improper rotation (S''n''). The identity operation (E) consists of leaving the molecule as it is. This is equivalent to any number of full rotations around any axis. This is a symmetry of all molecules, whereas the symmetry group of a chiral molecule consists of only the identity operation. An identity operation is a characteristic of every molecule even if it has no symmetry. Rotation around an axis (C''n'') consists of rotating the molecule around a specific axis by a specific angle. It is rotation through the angle 360°/''n'', where ''n'' is an integer, about a rotation axis. For example, if a water molecule rotates 180° around the axis that passes through the oxygen atom and between the hydrogen atoms, it is in the same configuration as it started. In this case, , since applying it twice produces the identity operation. In molecules with more than one rotation axis, the Cn axis having the largest value of n is the highest order rotation axis or principal axis. For example in boron trifluoride (BF3), the highest order of rotation axis is C3, so the principal axis of rotation is C3. In the reflection operation (σ) many molecules have mirror planes, although they may not be obvious. The reflection operation exchanges left and right, as if each point had moved perpendicularly through the plane to a position exactly as far from the plane as when it started. When the plane is perpendicular to the principal axis of rotation, it is called σ''h'' (horizontal). Other planes, which contain the principal axis of rotation, are labeled vertical (σ''v'') or dihedral (σ''d''). Inversion (i ) is a more complex operation. Each point moves through the center of the molecule to a position opposite the original position and as far from the central point as where it started. Many molecules that seem at first glance to have an inversion center do not; for example, methane and other Tetrahedron, tetrahedral molecules lack inversion symmetry. To see this, hold a methane model with two hydrogen atoms in the vertical plane on the right and two hydrogen atoms in the horizontal plane on the left. Inversion results in two hydrogen atoms in the horizontal plane on the right and two hydrogen atoms in the vertical plane on the left. Inversion is therefore not a symmetry operation of methane, because the orientation of the molecule following the inversion operation differs from the original orientation. And the last operation is improper rotation or rotation reflection operation (S''n'') requires rotation of  360°/''n'', followed by reflection through a plane perpendicular to the axis of rotation.

## Cryptography

Very large groups of prime order constructed in elliptic curve cryptography serve for public-key cryptography. Cryptographical methods of this kind benefit from the flexibility of the geometric objects, hence their group structures, together with the complicated structure of these groups, which make the discrete logarithm very hard to calculate. One of the earliest encryption protocols, Caesar cipher, Caesar's cipher, may also be interpreted as a (very easy) group operation. Most cryptographic schemes use groups in some way. In particular Diffie–Hellman key exchange uses finite cyclic groups. So the term group-based cryptography refers mostly to cryptographic protocols that use infinite nonabelian groups such as a braid group.

*List of group theory topics *Examples of groups

# References

* * * * * Shows the advantage of generalising from group to groupoid. * An introductory undergraduate text in the spirit of texts by Gallian or Herstein, covering groups, rings, integral domains, fields and Galois theory. Free downloadable PDF with open-source GNU Free Documentation License, GFDL license. * * * Conveys the practical value of group theory by explaining how it points to symmetries in
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
and other sciences. * * Mark Ronan, Ronan M., 2006. ''Symmetry and the Monster''. Oxford University Press. . For lay readers. Describes the quest to find the basic building blocks for finite groups. * A standard contemporary reference. * * Inexpensive and fairly readable, but somewhat dated in emphasis, style, and notation. * *