Group theory

In mathematics, group theory is the field that studies the algebraic structures known as groups.

Groups were first introduced in mathematics in the 19th century by mathematicians like Galois, Abel and Cauchy in their quest for general solutions of polynomial equations. The formal abstract definition—as described in the group article—was, however, not introduced until the 20th century.

Groups have become a central object in the study of abstract algebra, and are building blocks of more elaborate algebraic structures such as rings, fields, and vector spaces, and recur throughout mathematics. Group theory has many applications in physics and chemistry, and is potentially applicable in any situation characterized by symmetry.

The classification of finite simple groups is a major mathematical achievement of the 20th century.

History
There are three historical roots of group theory: the theory of algebraic equations, number theory and geometry. Euler, Gauss, Lagrange, Abel and French mathematician Galois were early researchers in the field of group theory. Galois is honored as the first mathematician linking group theory and field theory, with the theory that is now called Galois theory.

An early source occurs in the problem of forming an $$m$$th-degree equation having as its roots m of the roots of a given $$n$$th-degree equation ($$m < n$$). For simple cases the problem goes back to Hudde (1659). Saunderson (1740) noted that the determination of the quadratic factors of a biquadratic expression necessarily leads to a sextic equation, and Le Sœur (1748) and Waring (1762 to 1782) still further elaborated the idea.

A common foundation for the theory of equations on the basis of the group of permutations was found by mathematician Lagrange (1770, 1771), and on this was built the theory of substitutions. He discovered that the roots of all resolvents (résolvantes, réduites) which he examined are rational functions of the roots of the respective equations. To study the properties of these functions he invented a Calcul des Combinaisons. The contemporary work of Vandermonde (1770) also foreshadowed the coming theory.

Ruffini (1799) attempted a proof of the impossibility of solving the quintic and higher equations. Ruffini distinguished what are now called intransitive and transitive, and imprimitive and primitive groups, and (1801) uses the group of an equation under the name l'assieme delle permutazioni. He also published a letter from Abbati to himself, in which the group idea is prominent.

Galois found that if $$r_1, r_2, \ldots, r_n$$ are the $$n$$ roots of an equation, there is always a group of permutations of the $$r$$'s such that (1) every function of the roots invariable by the substitutions of the group is rationally known, and (2), conversely, every rationally determinable function of the roots is invariant under the substitutions of the group. Galois also contributed to the theory of modular equations and to that of elliptic functions. His first publication on group theory was made at the age of eighteen (1829), but his contributions attracted little attention until the publication of his collected papers in 1846 (Liouville, Vol. XI).

Arthur Cayley and Augustin Louis Cauchy were among the first to appreciate the importance of the theory, and to the latter especially are due a number of important theorems. The subject was popularized by Serret, who devoted section IV of his algebra to the theory; by Camille Jordan, whose Traité des Substitutions is a classic; and to Eugen Netto (1882), whose Theory of Substitutions and its Applications to Algebra was translated into English by Cole (1892). Other group theorists of the nineteenth century were Bertrand, Charles Hermite, Frobenius, Leopold Kronecker, and Emile Mathieu.

Walther von Dyck was the first (in 1882) to define a group in the full abstract sense of this entry.

The study of what are now called Lie groups, and their discrete subgroups, as transformation groups, started systematically in 1884 with Sophus Lie; followed by work of Killing, Study, Schur, Maurer, and Cartan. The discontinuous (discrete group) theory was built up by Felix Klein, Lie, Poincaré, and Charles Émile Picard, in connection in particular with modular forms and monodromy.

The classification of finite simple groups is a vast body of work from the mid 20th century, classifying all the finite simple groups.

Other important contributors to group theory include Emil Artin, Emmy Noether, Sylow, and many others.

Alfred Tarski proved elementary group theory undecidable.

University of Florida Graduate Research Professor John Griggs Thompson and European mathematician Jacques Tits won the 2008 Abel Prize for their contributions to group theory.

Basic definitions
A group (G, •) is a set G closed under a binary operation • satisfying the following 3 axioms:


 * Associativity: For all a, b and c in G, (a • b) • c = a • (b • c).
 * Identity element: There exists an e&isin;G such that for all a in G, e • a = a • e = a.
 * Inverse element: For each a in G, there is an element b in G such that a • b = b • a = e, where e is an identity element.

Basic examples for groups are the integers Z with addition operation, or rational numbers without zero Q\{0} with multiplication. More generally, for any ring R, the units in R form a multiplicative group. See the group article for an illustration of this definition and for further examples. Groups include, however, much more general structures than the above. Group theory is concerned with proving abstract statements about groups, regardless of the actual nature of element and the operation of the groups in question.

A subset H ⊂ G is a subgroup if the restriction of • to H is a group operation on H. It is called normal, if left and right cosets agree, i.e. gH = Hg for all g in G. Normal subgroups play a distinguished role by virtue of the fact that the collection of cosets of a normal subgroup N in a group G naturally inherits a group structure, enabling the formation of the quotient group, usually denoted G/N (also called a factor group). The Butterfly lemma is a technical result on the lattice of subgroups of a group.

A group homomorphism is a map f : G &rarr; H between two groups that preserves the structure imposed by the operation, i.e.
 * f(a•b) = f(a) • f(b).

Bijective (in-, surjective) maps are isomorphisms of groups (mono-, epimorphisms, respectively). The kernel ker(f) is always a normal subgroup of the group. For f as above, the fundamental theorem on homomorphisms relates the structure of G and H, and of the kernel and image of the homomorphism, namely
 * G / ker(f) &cong; im(f)

Groups together with group homomorphisms form a category.

Finiteness conditions
The order |G| (or o(G)) of a group is the cardinality of G. If the |G| is (in-)finite, then G itself is called (in-)finite. An important class is the group of permutations or symmetric groups of N letters, denoted SN. Cayley's theorem exhibits any finite group G as a subgroup of the symmetric group on G. The theory of finite groups is very rich. Lagrange's theorem states that the order of any subgroup H of a finite group G divides the order of G. A partial converse is given by the Sylow theorems: if pn is the greatest power of a prime p dividing the order of a finite group G, then there exists a subgroup of order pn, and the number of these subgroups is also known.

More generally, certain conditions on chains of subgroups, parallel to the notion of Noetherian and Artinian rings, allow to deduce further properties. For example the Krull-Schmidt theorem states that a group satisfying certain finiteness conditions for chains of its subgroups, can be uniquely written as a finite direct product of indecomposable subgroups.

Another, yet slightly weaker, level of finiteness is the following: a subset A of G is said to generate the group if any element h can be written as the product of elements of A. A group is said to be finitely generated if it is possible to find a finite subset A generating the group. Finitely generated groups are in many respects as well-treatable as finite groups.

Abelian groups
The category of groups can be subdivided in several ways. A particularly well-understood class of groups are the so-called abelian (in honor of Niels Abel, or commutative) groups, i.e. the ones satisfying
 * a • b = b • a for all a, b in G.

Another way of saying this is that the commutator
 * [a, b] := a&minus;1b&minus;1ab

equals the identity element. A non-abelian group is a group that is not abelian. Even more particular, cyclic groups are the groups generated by a single element. Being either isomorphic to Z or to Zn, the integers modulo n, they are always abelian. Any finitely generated abelian group is known to be a direct sum of groups of these two types. The category of abelian groups is an abelian category. In fact, abelian groups serve as the prototype of abelian categories. A converse is given by Mitchell's embedding theorem.

Normal series
Most of the notions developed in group theory are designed to tackle non-abelian groups. Given a group G and a normal subgroup N of G, denoted N ⊲ G, there is an exact sequence:
 * 1 &rarr; N &rarr; G &rarr; H &rarr; 1,

where 1 denotes the trivial group and H is the quotient G/N. This permits the decomposition of G into two smaller pieces. The other way round, given two groups N and H, a group G fitting into an exact sequence as above is called an extension of H by N. Given H and N there are many different group extensions G, which leads to the extension problem. There is always at least one extension, called the trivial extension, namely the direct sum G = N ⊕ H, but usually there are more. For example, the Klein four-group is a non-trivial extension of Z2 by Z2. This is a first glimpse of homological algebra and Ext functors.

Many properties for groups, for example being a finite group or a p-group are stable under extensions and sub- and quotient groups, i.e. if N and H have the property, then so does G and vice versa. Therefore much information about the group is preserved while breaking it into pieces by means of exact sequences. If this process has come to an end, i.e. if a group G does not have any (non-trivial) normal subgroups, G is called simple.

The name simple can be slightly misleading insofar that the group does not have to be easy to understand. The following concepts remedy this problem: repeatedly taking normal subgroups (if they exist) leads to normal series:
 * 1 = G0 ⊲ G1 ⊲ ... ⊲ Gn = G,

i.e. any Gi is a normal subgroup of the next one Gi+1. A group is solvable (or soluble) if it has a normal series all of whose quotients are abelian. Imposing further commutativity constraints on the quotients Gi+1 / Gi, one obtains central series which lead to nilpotent groups. They are a close approximation of abelian groups in the sense that
 * [...[[g1, g2], g3] ..., gn]=1

for all choices of group elements gi.

There may be distinct normal series for a group G. If it is impossible to refine a given series by inserting further normal subgroups, it is called composition series. By the Jordan-Hölder theorem any two composition series of a given group are equivalent.

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 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 {gi}i &isin; 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 〈F | D &rang;. For example, the group Z = 〈a | &rang; can be generated by one element a (equal to +1 or &minus;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. It is particularly useful where finitness 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&rang;

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. 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
 * &rho; : G &rarr; 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 &rho;(g) such that &rho;(g) ∘ &rho;(h) = &rho;(gh) 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 &rho;, 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. 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 L2-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
 * 1) 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.
 * 2) 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.
 * 3) If instead angles are preserved, one speaks of conformal maps. Conformal maps give rise to Kleinian groups, for example.
 * 4) Symmetries are not restricted to geometrical objects, but include algebraic objects as well: the equation
 * x4 &minus; 7x2 + 12 = 0
 * has the solutions +2, &minus;2, $$+\sqrt{3}$$, and $$-\sqrt{3}$$. Exchanging &minus;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, S5, 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 likewise uses group theory in many ways. The theory of abelian varieties, that is projective varieties with a group structure is particularly ample. The presence of a group operation yields additional information which makes these varieties particularly accessible. They also often serve as a test for new conjectures. The one-dimensional case, namely elliptic curves are studied in particular detail. Very large groups of prime order constructed in Elliptic-Curve Cryptography serve for public key cryptography. 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.

Algebraic number theory would not exist without group theory. For example, Euler's product formula

\begin{align} \sum_{n\geq 1}\frac{1}{n^s}& = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}} \\ \end{align} \!$$ 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.


 * 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.

Miscellany
In philosophy, Ernst Cassirer related group theory to the theory of perception of Gestalt Psychology. He took the Perceptual Constancy of that psychology as analogous to the invariants of group theory.