List of group theory topics

Structures and operations

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

Basic properties of groups

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 Elementary group theory Euler's theorem Fitting subgroup Generalized Fitting subgroup Hamiltonian group Identity element

Lagrange's theorem Multiplicative inverse Normal subgroup perfect group p-core Schreier refinement theorem Subgroup Transversal Torsion subgroup Zassenhaus lemma

Group homomorphisms

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

Basic types of groups

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 Solvable group P-group Pro-finite group

Simple groups and their classification

• Classification of finite simple groups

Algebraic group Alternating group Borel subgroup Chevalley group Conway group Feit–Thompson theorem Fischer group General linear group Group of Lie type

Group scheme Janko group Lie group Simple Lie group List of finite simple groups Mathieu group Monster group Baby Monster group Bimonster Parabolic subgroup

Projective group Simple group Quasisimple group Special linear group Symmetric group Thompson group (finite) Tits group Weyl group

Permutation and symmetry groups

Arithmetic group Braid group Burnside's lemma 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 Homogeneous space Isometry group Orbit (group theory) Permutation Permutation group

Rubik's Cube Space group Stabilizer subgroup Steiner system Strong generating set Symmetry Symmetric group Symmetry group Wallpaper group

Concepts groups share with other mathematics

• Associativity
• Bijection
• Bilinear operator
• Binary operation
• Commutative
• Congruence relation
• Equivalence class
• Equivalence relation
• Lattice (group)
• Lattice (discrete subgroup)
• Multiplication table
• Prime number
• Up to

Mathematical objects making use of a group operation

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 Modular arithmetic

Number Pauli matrices Real number Quaternion Quaternion group Tensor

Mathematical fields & topics making important use of group theory

• Algebraic geometry
• Algebraic topology
• Discrete space
• Fundamental group
• Geometry
• Homology
• Minkowski's theorem
• Topological group

Algebraic structures related to groups

Field Finite field Galois theory Group ring Group with operators Heap Linear algebra Magma

Module Monoid Monoid ring Quandle Quasigroup Ring Semigroup Vector space

Group representations

See also: List of representation theory representation theory topics and 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

• 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

Famous problems

• Burnside's problem
• Classification of finite simple groups
• Herzog-Schönheim conjecture
• Subset sum problem
• Whitehead problem
• Word problem for groups

Other topics

• Amenable group
• Capable group
• Commensurability (mathematics)
• Compact group
• Compactly generated group
• Complete group
• Congruence subgroup
• Continuous symmetry
• Frattini subgroup
• Growth rate
• Heisenberg group, discrete Heisenberg group
• Molecular symmetry
• Nielsen transformation
• Tarski monster group
• Thompson groups
• Tietze transformation
• Transfer (group theory)

Group theorists

N. Abel M. Aschbacher R. Baer R. Brauer W. Burnside R. Carter A. Cauchy A. Cayley J.H. Conway R. Dedekind L.E. Dickson

M. Dunwoody W. Feit B. Fischer H. Fitting G. Frattini G. Frobenius E. Galois G. Glauberman D. Gorenstein R.L. Griess M. Hall, Jr.

P. Hall G. Higman D. Hilbert O. Hölder B. Huppert K. Iwasawa Z. Janko C. Jordan F. Klein A. Kurosh J.L. Lagrange

C. Leedham-Green F.W. Levi Sophus Lie W. Magnus E. Mathieu G.A. Miller B.H. Neumann H. Neumann J. Nielson Emmy Noether Ø. Ore

O. Schreier I. Schur R. Steinberg M. Suzuki L. Sylow J. Thompson J. Tits Helmut Wielandt H. Zassenhaus M. Zorn

See also

• List of abstract algebra topics
• List of category theory topics
• List of Lie group topics