Group theory

Group theory is a mathematical discipline, the part of abstract algebra that studies the algebraic structures known as groups. The development of group theory sprang from three main sources: number theory, theory of algebraic equations, and geometry. The number-theoretic strand was started by Leonhard Euler and taken up by Gauss, who developed modular arithmetic and considered 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. Galois coined the term “group” and established a connection between the nascent theory of groups and field theory, which is known as Galois theory. In geometry, groups first became important in projective geometry and, later, non-Euclidean geometry. Felix Klein in his Erlangen program famously proclaimed group theory to be the organizing principle behind the very meaning of geometry.

Groups manifest themselves as symmetry groups of various physical systems, such as crystals and the hydrogen atom. Thus group theory and the closely related representation theory have many applications in physics and chemistry.

The concept of a group is a central concept of abstract algebra: other algebraic structures, such as rings, fields, and vector spaces are elaborations of groups, which are endowed with additional operations. Groups recur throughout mathematics, and methods of group theory had a strong influence on ring theory and other parts of algebra. Linear algebraic groups and Lie groups are two classes of groups whose theory has been tremendously advanced, and became the subject areas of their own.

A central question of group theory throughout much of the last century was the classification of finite simple groups. The result of a collaborative effort mostly from 1960-1980 and totaling more than ten thousand pages, it is one of the most important mathematical achievements of the 20th century.

History

There are three historical roots of group theory: the theory of algebraic equations, number theory and geometry.

Historically, the first use of groups to determine the solvability of polynomial equations was done by Évariste Galois, in the 1830s. Investigations were pushed further, mainly in the guise of permutation groups, by Arthur Cayley and Augustin Louis Cauchy. 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 (first implicitly and later explicitly) used in algebraic number theory.

The different scope of these early sources resulted in different notions of groups. The theory of groups was unified starting around 1880. Since, the impact of group theory has been ever growing, giving rise to the birth of abstract algebra in the early 20th century, representation theory, and many more influential spin-off domains. The classification of finite simple groups is a vast body of work from the mid 20th century, classifying all the finite simple groups.

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 by generators and relations.

Permutation groups

The first class of groups to undergo a systematic study was permutation groups. Given any set "X" and a collection "G" of bijections of "X" into itself (known as "permutations") that is closed under compositions and inverses, "G" is a group acting on "X". If "X" consists of "n" elements and "G" consists of "all" permutations, "G" is the symmetric group "S"_"n"; in general, "G" is a subgroup of the symmetric group of "X". An early construction due to Cayley exhibited any group as a permutation group, acting on itself ("X" = "G") 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 "n" ≥ 5, 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 "n" ≥ 5 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 transformations. This action makes matrix groups conceptually similar to permutation groups, and geometry of the action may be usefully expoited to establish properties of the group "G".

Transformation groups

Permutation groups and matrix groups are special cases of transformation groups: 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. The concept of a transformation group is closely related with the concept of a symmetry group: transformation groups frequently consist of "all" transformations that preserve a certain structure. The theory of transformation groups forms a bridge connecting group theory with differential geometry. A long line of research, originating with Lie and Klein, considers group actions on manifolds by 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 by "generators and relations",

: $G\; =\; langle\; S|R\; angle.$

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. 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, as well as the classes of group with a given such property: finite groups, periodic groups, simple groups, 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 the works of Hilbert, Emil Artin, Emmy Noether, and mathematicians of their school.

Topological and algebraic groups

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),

: $m:\; G\; imes\; G\; o\; G,\; (g,h)mapsto\; gh,\; quad\; i:G\; o\; G,\; gmapsto\; g^\{-1\},$

are compatible with this structure, i.e. are continuous, smooth or regular (in the sense of algebraic geometry) maps then "G" becomes a topological group, a Lie group, or an algebraic group. [This process of imposing extra structure has been formalized through the notion of a group object in a suitable category. Thus Lie groups are group objects in the category of differentiable manifolds and affine algebraic groups are group objects in the category of affine algebraic varieties.]

The presence of extra structure relates these types of groups with other mathematical disciplines and means that more tools are available in their study. Topological groups form a natural domain for abstract harmonic analysis, whereas Lie groups (frequently realized as transformation groups) are the mainstays of differential geometry and unitary representation theory. 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.

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 nowrap|"g" • "h". A more important way of defining a group is by "generators and relations", also called the "presentation" of a group. Given any set "F" of generators {"g"_"i"}_{"i" ∈ "I"}, the free group generated by "F" surjects onto the group "G". The kernel of this map is called subgroup of relations, generated by some subset "D". The presentation is usually denoted by nowrap begin〈"F" | "D" 〉nowrap end. For example, the group nowrap begin

Z = 〈"a" | 〉nowrap end can be generated by one element "a" (equal to +1 or −1) and no relations, because "n"·1 never equals 0 unless "n" is zero. A string consisting of generator symbols is called a "word".

Combinatorial group theory studies groups from the perspective of generators and relations. [harvnb|Schupp|Lyndon|2001] 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 groups. For example, one can show that every subgroup of a free group is free.

There are several natural questions arising from giving a group by its presentation. The "word problem" asks whether two words are effectively the same group element. By relating the problem to Turing machines, one can show that there is in general no algorithm solving this task. An equally difficult problem is, whether two groups given by different presentations are actually isomorphic. For example Z can also be presented by:〈"x", "y" | "xyxyx" = 1⟩and it is not obvious (but true) that this presentation is isomorphic to the standard one above.

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. [harvnb|La Harpe|2000] 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 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-isometric (i.e. looks similar from the far) to the space "X".

Representation of groups

Saying that a group "G" "acts" on a set "X" means that every element defines a bijective map on a set 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 "V" is a group homomorphism:"ρ" : "G" → "GL"("V"),where "GL"("V") consists of the invertible linear transformations of "V". In other words, to every group element "g" is assigned an automorphism "ρ"("g") such that nowrap begin"ρ"("g") ∘ "ρ"("h") = "ρ"("gh")nowrap end for any "h" in "G".

This definition can be understood in two directions, both of which give rise to whole new domains of mathematics. [Such as group cohomology or equivariant K-theory.] 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. [In particular, if the representation is faithful.] On the other hand, given a well-understood group acting on a complicated object, this simplifies the study of the object in question. For example, if "G" is finite, it is known that "V" above decomposes into irreducible parts. These parts in turn are much more easily manageable than the whole "V" (via Schur's lemma).

Given a group "G", representation theory 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 "1", acting on the "L"²-space of periodic functions.

Connection of groups and symmetry

Given a structured object "X" of any sort, a symmetry is a mapping of the object onto itself which preserves the structure. This occurs in many cases, for example
#If "X" is a set with no additional structure, a symmetry is a bijective 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 structure or any other metric space, a symmetry is a bijection 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: the equation::"x"⁴ − 7"x"² + 12 = 0:has the solutions +2, −2, $+sqrt\{3\}$, and $-sqrt\{3\}$. Exchanging −2 and +2 and the two square roots determines a group, the Galois group belonging to the equation.

The axioms of a group formalize the essential aspects of symmetry. Symmetries form a group: they are closed because if you take a symmetry of an object, and then apply another symmetry, the result will still be a symmetry. The identity keeping the object fixed is always a symmetry of an object. Existence of inverses is guaranteed by the undoing the symmetry and the associativity comes from the fact that symmetries are functions on a space, and composition of functions are associative.

Frucht's theorem says that every group is the symmetry group of some 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. 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 are special cases of groups. 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 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 is another domain which prominently associates groups to the objects the theory is interested in. There, groups are used to describe certain invariants of topological 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 deformation. For example, the fundamental group "counts" how many paths in the space are essentially different. The Poincaré conjecture, proved in 2002/2003 by 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 stakes in a crucial 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 and cryptography likewise uses group theory in many ways. 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. [See the Birch-Swinnerton-Dyer conjecture, one of the millennium problems] 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's cipher, may also be interpreted of a (very easy) group operation. In another direction, toric varieties are algebraic varieties acted on by a torus. Toroidal embeddings have recently led to advances in algebraic geometry, in particular resolution of singularities. [Citation | last1=Abramovich | first1=Dan | last2=Karu | first2=Kalle | last3=Matsuki | first3=Kenji | last4=Wlodarczyk | first4=Jaroslaw | title=Torification and factorization of birational maps | id=MathSciNet | id = 1896232 | year=2002 | journal=Journal of the American Mathematical Society | issn=0894-0347 | volume=15 | issue=3 | pages=531–572]

Algebraic number theory is a special case of group theory, thereby following the rules of the latter. For example, Euler's product formula:captures the fact that any integer decomposes in a unique way into primes. The failure of this statement for more general rings gives rise to class groups and regular primes, which feature in Kummer's treatment of Fermat's Last Theorem.

*The concept of the Lie group (named after mathematician Sophus Lie) is important in the study of differential equations and manifolds; they describe the symmetries of continuous geometric and analytical structures. Analysis on these and 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. [Citation | last1=Lenz | first1=Reiner | title=Group theoretical methods in image processing | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Computer Science | isbn=978-0-387-52290-6 | year=1990 | volume=413]

*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.

*The presence of the 12-periodicity in the circle of fifths yields applications of elementary group theory in musical set theory.

*An understanding of group theory is also important in physics and chemistry and material science. In physics, groups are important because they describe the symmetries which the laws of physics seem to obey. 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: Standard Model, Gauge theory, Lorentz group, Poincaré group

*In chemistry, groups are used to classify crystal structures, regular polyhedra, and the symmetries of molecules. The assigned point groups can then be used to determine physical properties (such as polarity and chirality), spectroscopic properties (particularly useful for Raman spectroscopy and Infrared spectroscopy), and to construct molecular orbitals.

See also

*Group (mathematics)
*Glossary of group theory
*List of group theory topics

Notes

References

* | year=1991 | volume=126
* | year=1969 | journal=Communications of the Association for Computing Machinery | issn=0001-0782 | volume=12 | pages=3–12
*Connell, Edwin, " [http://www.math.miami.edu/~ec/book/ Elements of Abstract and Linear Algebra.] " Free online textbook.
*
* | year=1986 | journal=Mathematics Magazine | issn=0025-570X | volume=59 | issue=4 | pages=195–215
*
*cite book | author=Livio, M. | title= The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry | publisher=Simon & Schuster | year=2005 | id=ISBN 0-7432-5820-7 A pleasant read, explaining the importance of group theory and how its symmetries point to symmetries in physics and other sciences. Conveys well the practical value of group theory.
*
*cite book | author=Rotman, Joseph | title=An introduction to the theory of groups | location=New York | publisher=Springer-Verlag | year=1994 | id=ISBN 0-387-94285-8 A standard contemporary reference.
* | year=1994
*
*cite book | author=Scott, W. R. | title= Group Theory | location=New York | publisher=Dover | year=1987 | origyear=1964 | id=ISBN 0-486-65377-3 Inexpensive and fairly readable, but somewhat dated in emphasis, style, and notation.
* | year=1972
* |journal= Bull. Amer. Math. Soc. (N.S.) | issn =0273-0979 |volume=43 | year= 2006 | pages=305--364 This shows an advantage of the generalisation from group to groupoid.

External links

* [http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Abstract_groups.html History of the abstract group concept]
* [http://www.bangor.ac.uk/r.brown/hdaweb2.htm Higher dimensional group theory] This presents a view of group theory as level one of a theory which extends in all dimensions, and has applications in homotopy theory and to higher dimensional nonabelian methods for local-to-global problems.