Group (mathematics)

Group (mathematics)
Name	Group
Caption	Symmetry group of an equilateral triangle
Field	Mathematics
Introduced	Évariste Galois
Examples	Cyclic group, Symmetric group, Alternating group, Dihedral group

Group (mathematics) A group is an algebraic structure consisting of a set equipped with an operation that combines any two elements to form a third, satisfying associativity, an identity element, and inverses. Groups arise in the work of Évariste Galois, Arthur Cayley, Augustin-Louis Cauchy, Niels Henrik Abel and appear throughout studies by Emmy Noether, Henri Poincaré, Sophus Lie and Felix Klein. The concept is central to areas influenced by Isaac Newton, Carl Friedrich Gauss, David Hilbert and Emmy Noether and underpins theories developed at institutions like École Normale Supérieure, University of Göttingen, Princeton University and University of Cambridge.

Definition and basic properties

A group is defined by a set G and a binary operation · such that for all a, b, c in G the operation is associative; there exists an identity element e with e·a = a·e = a; and every a has an inverse a^{-1} with a·a^{-1} = a^{-1}·a = e. Foundational work by Galois, Cauchy, Cayley and Noether formalized axioms used in texts from David Hilbert to modern accounts at Massachusetts Institute of Technology and University of Chicago. Basic properties include cancellation laws, uniqueness of identity and inverses, and consequences like the order of an element dividing group order in finite contexts, a theme in results discussed by Lagrange and studied by Évariste Galois and William Rowan Hamilton.

Examples and classes of groups

Concrete examples include finite groups such as the Symmetric group S_n studied by Augustin-Louis Cauchy and Camille Jordan, the Alternating group A_n appearing in work by Émile Mathieu and Évariste Galois, and the Dihedral group D_n modeling polygonal symmetries examined by Felix Klein and Sophus Lie. Infinite examples include Cyclic groups like Z, additive groups of fields studied by Carl Friedrich Gauss, matrix groups like General linear group GL(n) and Special linear group SL(n) relevant to Hermann Weyl and Élie Cartan, and continuous Lie groups such as SO(n), SU(n), Sp(n) central to work by Sophus Lie, Élie Cartan, Hermann Weyl and Eugene Wigner. Other classes include Abelian groups investigated by Niels Henrik Abel and Emmy Noether, Solvable groups appearing in solvability of polynomials from Galois and Burnside studies, Simple groups culminating in the classification by groups like the Monster group discovered through work at Cambridge and Institute for Advanced Study, and p-groups central to research by Philip Hall and John Thompson.

Subgroups and homomorphisms

A subgroup is a subset closed under the group operation and inverses; normal subgroups introduced in Évariste Galois's theory permit quotient constructions. Homomorphisms preserve structure and give kernels and images, leading to the isomorphism theorems developed by Noether and Emmy Noether's collaborators at University of Göttingen. Important subgroup types include Sylow subgroups linked to Peter Ludwig Sylow's theorems, Hall subgroups from Philip Hall, maximal subgroups studied by Wolfgang Gaschütz and subnormal subgroups appearing in work by Bertrand Russell and Otto Schreier. Concepts like centralizers, normalizers, center and derived subgroup connect to investigations by William Burnside, Isaacs and John Thompson.

Group actions and applications

A group action is a homomorphism from a group to permutations of a set; actions encode symmetry studied by Felix Klein in his Erlangen program and by Élie Cartan in differential geometry. Applications span crystallography at Max von Laue and William H. Bragg, particle physics via Eugene Wigner and Murray Gell-Mann using Lie group representations, coding theory at Claude Shannon and Richard Hamming, combinatorics by Paul Erdős and Ronald Graham, and topology through fundamental groups in work by Henri Poincaré and Hurewicz. Group actions underpin model-building at CERN and symmetry classification in chemistry associated with Linus Pauling.

Structure theorems and classification

Structure theorems describe composition series, Jordan–Hölder theorems named for Camille Jordan and Otto Hölder, and the classification of finite simple groups achieved by collaborative efforts involving Daniel Gorenstein, Robert Guralnick, Michael Aschbacher, John Conway and others at institutions like University of Cambridge and Rutgers University. The Sylow theorems from Peter Ludwig Sylow give p-subgroup structure; the Feit–Thompson theorem by Walter Feit and John Thompson handles odd-order groups; the classification yields families like Chevalley groups, Ree groups, Suzuki groups and sporadic groups including the Monster group explored by John Conway and Simon Norton. For infinite groups, decomposition results such as the Krull–Schmidt theorem and Bass–Serre theory developed by Hyman Bass and Jean-Pierre Serre describe amalgams and fundamental group actions on trees.

Representations and extensions

Representation theory studies homomorphisms from groups to general linear groups, with foundational contributions by Ferdinand Georg Frobenius, Issai Schur, Hermann Weyl and later work at Institute for Advanced Study by Hugh Conway and George Mackey. Maschke's theorem and character theory analyze representations of finite groups in the tradition of Frobenius and Burnside; Mackey and Harish-Chandra advanced harmonic analysis on groups. Group extensions, cohomology of groups introduced by Samuel Eilenberg and Norman Steenrod and developed by Jean-Pierre Serre, classify ways groups combine and produce obstruction classes studied by Shapiro and Eckmann. Modern interactions connect representations to number theory in the Langlands program led by Robert Langlands and to quantum groups investigated by Vladimir Drinfeld and Michio Jimbo.

Category:Algebra