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List of group theory topics
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{{Short description|None}} {{Group theory sidebar |Basics}} In [[mathematics]] and [[abstract algebra]], ''[[group theory]]'' studies the [[algebraic structure]]s known as [[group (mathematics)|groups]]. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as [[ring (mathematics)|rings]], [[field (mathematics)|fields]], and [[vector space]]s, can all be seen as groups endowed with additional [[operation (mathematics)|operation]]s and [[axiom]]s. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. [[Linear algebraic group]]s and [[Lie group]]s 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]]s and the [[hydrogen atom]], may be modelled by [[symmetry group]]s. Thus group theory and the closely related [[representation theory]] have many important applications in [[physics]], [[chemistry]], and [[materials science]]. Group theory is also central to [[public key cryptography]]. ==Structures and operations== {{colbegin|colwidth=22em}} * [[Group extension#Central extension|Central extension]] * [[Direct product of groups]] * [[Direct sum of groups]] * [[Extension problem]] * [[Free abelian group]] * [[Free group]] * [[Free product]] * [[Generating set of a group]] * [[Group cohomology]] * [[Group extension]] * [[Presentation of a group]] * [[Product of group subsets]] * [[Schur multiplier]] * [[Semidirect product]] * [[Sylow theorems]] ** [[Hall subgroup]] * [[Wreath product]] {{colend}} ==Basic properties of groups== {{colbegin|colwidth=22em}} *[[Butterfly lemma]] * [[Center of a group]] * [[Centralizer and normalizer]] * [[Characteristic subgroup]] * [[Commutator]] * [[Composition series]] * [[Conjugacy class]] * [[Conjugate closure]] * [[Conjugation of isometries in Euclidean space]] * [[Core (group)]] * [[Coset]] * [[Derived group]] * [[Euler's theorem]] * [[Fitting subgroup]] **[[Generalized Fitting subgroup]] * [[Hamiltonian group]] * [[Identity element]] * [[Lagrange's theorem (group theory)|Lagrange's theorem]] * [[Multiplicative inverse]] * [[Normal subgroup]] * [[Perfect group]] * [[p-core]] * [[Schreier refinement theorem]] * [[Subgroup]] * [[Transversal (combinatorics)]] * [[Torsion subgroup]] * [[Zassenhaus lemma]] {{colend}} ===Group homomorphisms=== {{colbegin|colwidth=22em}} * [[Automorphism]] * [[Automorphism group]] * [[Factor group]] * [[Fundamental theorem on homomorphisms]] * [[Group homomorphism]] * [[Group isomorphism]] * [[Homomorphism]] * [[Isomorphism theorem]] * [[Inner automorphism]] * [[Order automorphism]] * [[Outer automorphism group]] * [[Quotient group]] {{colend}} ==Basic types of groups== {{colbegin|colwidth=22em}} * [[Examples of groups]] * [[Abelian group]] * [[Cyclic group]] ** [[Rank of an abelian group]] * [[Dicyclic group]] * [[Dihedral group]] * [[Divisible group]] * [[Finitely generated abelian group]] * [[Group representation]] * [[Klein four-group]] * [[List of small groups]] * [[Locally cyclic group]] * [[Nilpotent group]] * [[Non-abelian group]] * [[Solvable group]] * [[P-group]] * [[Pro-finite group]] {{colend}} ===Simple groups and their classification=== *[[Classification of finite simple groups]] {{colbegin|colwidth=22em}} * [[Alternating group]] * [[Borel subgroup]] * [[Chevalley group]] * [[Conway group]] * [[Feit–Thompson theorem]] * [[Fischer group]] * [[General linear group]] * [[Group of Lie type]] * [[Group scheme]] * [[HN group]] * [[Janko group]] * [[Lie group]] ** [[Simple Lie group]] * [[Linear algebraic group]] * [[List of finite simple groups]] * [[Mathieu group]] * [[Monster group]] ** [[Baby Monster group]] ** [[Bimonster]] * [[Projective group]] * [[Reductive group]] * [[Simple group]] ** [[Quasisimple group]] * [[Special linear group]] * [[Symmetric group]] * [[Thompson group (finite)]] * [[Tits group]] * [[Weyl group]] {{colend}} ===Permutation and symmetry groups=== {{colbegin|colwidth=22em}} * [[Arithmetic group]] * [[Braid group]] * [[Burnside's lemma]] * [[Cayley's Theorem|Cayley's theorem]] * [[Coxeter group]] * [[Crystallographic group]] * [[Crystallographic point group]], [[Schoenflies notation]] * [[Discrete group]] * [[Euclidean group]] * [[Even and odd permutations]] * [[Frieze group]] * [[Frobenius group]] * [[Fuchsian group]] * [[Geometric group theory]] * [[Group action (mathematics)|Group action]] * [[Homogeneous space]] * [[Hyperbolic group]] * [[Isometry group]] * [[Orbit (group theory)]] * [[Permutation]] * [[Permutation group]] * [[Rubik's Cube group]] * [[Space group]] * [[Stabilizer subgroup]] * [[Steiner system]] * [[Strong generating set]] * [[Symmetry]] * [[Symmetric group]] * [[Symmetry group]] * [[Wallpaper group]] {{colend}} ==Concepts groups share with other mathematics== {{colbegin|colwidth=22em}} * [[Associativity]] * [[Bijection]] * [[Bilinear operator]] * [[Binary operation]] * [[Commutative]] * [[Congruence relation]] * [[Equivalence class]] * [[Equivalence relation]] * [[Lattice (group)]] * [[Lattice (discrete subgroup)]] * [[Multiplication table]] * [[Prime number]] * [[Up to]] {{colend}} ==Mathematical objects making use of a group operation== {{colbegin|colwidth=22em}} * [[Abelian variety]] * [[Algebraic group]] * [[Banach–Tarski paradox]] * [[Category of groups]] * [[Dimensional analysis]] * [[Elliptic curve]] * [[Galois group]] * [[Gell-Mann matrices]] * [[Group object]] * [[Hilbert space]] * [[Integer]] * [[Lie group]] * [[matrix (mathematics)|Matrix]] * [[Modular arithmetic]] * [[Number]] * [[Pauli matrices]] * [[Real number]] * [[Quaternion]] ** [[Quaternion group]] * [[Tensor]] {{colend}} ==Mathematical fields and topics making important use of group theory== * [[Algebraic geometry]] * [[Algebraic topology]] * [[Discrete space]] * [[Fundamental group]] * [[Geometry]] * [[Homology (mathematics)|Homology]] * [[Minkowski's theorem]] * [[Topological group]] ==Algebraic structures related to groups== {{colbegin|colwidth=22em}} * [[Field (mathematics)|Field]] * [[Finite field]] * [[Galois theory]] * [[Grothendieck group]] * [[Group ring]] * [[Group with operators]] * [[Heap (mathematics)|Heap]] * [[Linear algebra]] * [[Magma (algebra)|Magma]] * [[Module (mathematics)|Module]] * [[Monoid]] * [[Monoid ring]] * [[Quandle]] * [[Quasigroup]] * [[Quantum group]] * [[Ring (mathematics)|Ring]] * [[Semigroup]] * [[Vector space]] {{colend}} ==Group representations== {{see also|List of representation theory topics|List of harmonic analysis topics}} * [[Affine representation]] * [[Character theory]] * [[Great orthogonality theorem]] * [[Maschke's theorem]] * [[Monstrous moonshine]] * [[Projective representation]] * [[Representation theory]] * [[Schur's lemma]] ==Computational group theory== {{main|Computational group theory}} * [[Coset enumeration]] * [[Schreier's subgroup lemma]] * [[Schreier–Sims algorithm]] * [[Todd–Coxeter algorithm]] ==Applications== * [[Computer algebra system]] * [[Cryptography]] ** [[Discrete logarithm]] ** [[Triple DES]] ** [[Caesar cipher]] * [[Exponentiating by squaring]] * [[Knapsack problem]] * [[Shor's algorithm]] * [[Standard Model]] * [[Symmetry in physics]] ==Famous problems== * [[Burnside's problem]] * [[Classification of finite simple groups]] * [[Herzog–Schönheim conjecture]] * [[Subset sum problem]] * [[Whitehead problem]] * [[Word problem for groups]] ==Other topics== {{colbegin|colwidth=22em}} * [[Amenable group]] * [[Capable group]] * [[Commensurability (group theory)]] * [[Compact group]] * [[Compactly generated group]] * [[Complete group]] * [[Complex reflection group]] * [[Congruence subgroup]] * [[Continuous symmetry]] * [[Frattini subgroup]] * [[Growth rate (group theory)|Growth rate]] * [[Heisenberg group]], [[discrete Heisenberg group]] * [[Molecular symmetry]] * [[Nielsen transformation]] * [[Reflection group]] * [[Tarski monster group]] * [[Thompson groups]] * [[Tietze transformation]] * [[Transfer (group theory)]] {{colend}} ==Group theorists== {{colbegin|colwidth=15em}} * [[Niels Henrik Abel|N. Abel]] * [[Michael Aschbacher|M. Aschbacher]] * [[Reinhold Baer|R. Baer]] * [[Richard Brauer|R. Brauer]] * [[William Burnside|W. Burnside]] * [[Roger Carter (mathematician)|R. Carter]] * [[Augustin Louis Cauchy|A. Cauchy]] * [[Arthur Cayley|A. Cayley]] * [[John Horton Conway|J.H. Conway]] * [[Richard Dedekind|R. Dedekind]] * [[Leonard Eugene Dickson|L.E. Dickson]] * [[Martin Dunwoody|M. Dunwoody]] * [[Walter Feit|W. Feit]] * [[Bernd Fischer (mathematician)|B. Fischer]] * [[Hans Fitting|H. Fitting]] * [[Giovanni Frattini|G. Frattini]] * [[Ferdinand Georg Frobenius|G. Frobenius]] * [[Évariste Galois|E. Galois]] * [[George Glauberman|G. Glauberman]] * [[Daniel Gorenstein|D. Gorenstein]] * [[R. L. Griess|R.L. Griess]] * [[Marshall Hall (mathematician)|M. Hall, Jr.]] * [[Philip Hall|P. Hall]] * [[Graham Higman|G. Higman]] * [[David Hilbert|D. Hilbert]] * [[Otto Ludwig Hölder|O. Hölder]] * [[Bertram Huppert|B. Huppert]] * [[Kenkichi Iwasawa|K. Iwasawa]] * [[Zvonimir Janko|Z. Janko]] * [[Camille Jordan|C. Jordan]] * [[Felix Klein|F. Klein]] * [[Aleksandr Gennadievich Kurosh|A. Kurosh]] * [[Joseph-Louis Lagrange|J.L. Lagrange]] * [[Charles Leedham-Green|C. Leedham-Green]] * [[Friedrich Wilhelm Levi|F.W. Levi]] * [[Sophus Lie]] * [[Wilhelm Magnus|W. Magnus]] * [[Émile Léonard Mathieu|E. Mathieu]] * [[George Abram Miller|G.A. Miller]] * [[Bernhard Neumann|B.H. Neumann]] * [[Hanna Neumann|H. Neumann]] * [[Jakob Nielsen (mathematician)|J. Nielson]] * [[Emmy Noether]] * [[Øystein Ore|Ø. Ore]] * [[Otto Schreier|O. Schreier]] * [[Issai Schur|I. Schur]] * [[Robert Steinberg|R. Steinberg]] * [[Michio Suzuki (mathematician)|M. Suzuki]] * [[Peter Ludwig Mejdell Sylow|L. Sylow]] * [[John G. Thompson|J. Thompson]] * [[Jacques Tits|J. Tits]] * [[Helmut Wielandt]] * [[Hans Zassenhaus|H. Zassenhaus]] * [[Max August Zorn|M. Zorn]] {{colend}} ==See also== * [[List of abstract algebra topics]] * [[List of category theory topics]] * [[List of Lie group topics]] {{DEFAULTSORT:Group theory}} [[Category:Group theory|*]] [[Category:Properties of groups| ]] [[Category:Mathematics-related lists]] [[Category:Outlines of mathematics and logic]] [[Category:Outlines]]
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