Group (mathematics)

In mathematics, a group is a set equipped with a binary operation that combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, and help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study.Richard Borcherds in Mathematicians: An Outer View of the Inner World ^ a: Mathematical Reviews lists 3,224 research papers on group theory and its generalizations written in 2005.^ aa: The classification was announced in 1983, but gaps were found in the proof. See classification of finite simple groups for further information.^ b: The closure axiom is already implied by the condition that • be a binary operation. Some authors therefore omit this axiom. However, group constructions often start with an operation defined on a superset, so a closure step is common in proofs that a system is a group. Lang 2002^ c: See, for example, the books of Lang (2002, 2005) and Herstein (1996, 1975).^ d: However, a group is not determined by its lattice of subgroups. See Suzuki 1951.^ e: The fact that the group operation extends this canonically is an instance of a universal property.^ f: For example, if G is finite, then the size of any subgroup and any quotient group divides the size of G, according to Lagrange's theorem.^ g: The word homomorphism derives from Greek ὁμός—the same and μορφή—structure.^ h: The additive notation for elements of a cyclic group would be t • a, t in Z.^ i: See the Seifert–van Kampen theorem for an example.^ j: An example is group cohomology of a group which equals the singular cohomology of its classifying space.^ k: Elements which do have multiplicative inverses are called units, see Lang 2002, §II.1, p. 84.^ l: The transition from the integers to the rationals by adding fractions is generalized by the field of fractions.^ m: The same is true for any field F instead of Q. See Lang 2005, §III.1, p. 86.^ n: For example, a finite subgroup of the multiplicative group of a field is necessarily cyclic. See Lang 2002, Theorem IV.1.9. The notions of torsion of a module and simple algebras are other instances of this principle. ^ o: The stated property is a possible definition of prime numbers. See prime element.^ p: For example, the Diffie-Hellman protocol uses the discrete logarithm.^ q: The groups of order at most 2000 are known. Up to isomorphism, there are about 49 billion. See Besche, Eick & O'Brien 2001.^ r: The gap between the classification of simple groups and the one of all groups lies in the extension problem, a problem too hard to be solved in general. See Aschbacher 2004, p. 737.^ s: Equivalently, a nontrivial group is simple if its only quotient groups are the trivial group and the group itself. See Michler 2006, Carter 1989.^ t: More rigorously, every group is the symmetry group of some graph; see Frucht's theorem, Frucht 1939.^ u: More precisely, the monodromy action on the vector space of solutions of the differential equations is considered. See Kuga 1993, pp. 105–113.^ v: See Schwarzschild metric for an example where symmetry greatly reduces the complexity of physical systems.^ w: This was crucial to the classification of finite simple groups, for example. See Aschbacher 2004.^ x: See, for example, Schur's Lemma for the impact of a group action on simple modules. A more involved example is the action of an absolute Galois group on étale cohomology.^ y: Injective and surjective maps correspond to mono- and epimorphisms, respectively. They are interchanged when passing to the dual category. In mathematics, a group is a set equipped with a binary operation that combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, and help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study. Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; Poincaré groups, which are also Lie groups, can express the physical symmetry underlying special relativity; and point groups are used to help understand symmetry phenomena in molecular chemistry. The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. 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.a 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. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004.aa Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory. The modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial 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 (solutions). The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' 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 θn = 1 (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 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' 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 numbers. The convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques (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, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur'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 (from the late 1930s) and later by the work of Armand Borel and Jacques Tits. The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, 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 is ongoing to simplify the proof of this classification. These days, group theory is still a highly active mathematical branch, impacting many other fields.a One of the most familiar groups is the set of integers Z {displaystyle mathbb {Z} } which consists of the numbers