List of group theory topics

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This is a list of group theory topics, by Wikipedia page, to provide an overview of the topic and to allow those interested to use the "Related changes" feature to keep abreast of edits in this area.

See also: List of abstract algebra topics, List of category theory topics, list of Lie group topics.

Contents 1 Basic group structures and operations 2 Basic properties of groups 2.1 Group homomorphisms 3 Basic types of groups 3.1 Simple groups and their classification 3.2 Permutation and symmetry groups 4 Concepts groups share with other mathematics 5 Mathematical objects making use of a group operation 6 Mathematical fields & topics making important use of group theory 7 Algebraic structures related to groups 8 Group representations 9 Computational group theory 10 Applications 11 Famous problems 12 Other topics 13 Major contributors to group theory

[edit] Basic group structures and operations

Central extension Direct product Direct sum 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

[edit] 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 p-core Schreier refinement theorem Subgroup Transversal Torsion subgroup Zassenhaus lemma

[edit] 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

[edit] 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

[edit] 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 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

[edit] 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

[edit] 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

[edit] 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

[edit] Mathematical fields & topics making important use of group theory

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

[edit] Algebraic structures related to groups

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

Monoid Monoid ring Quandle Quasigroup Ring Semigroup Vector space

[edit] Group representations

See also list of harmonic analysis and representation theory topics

• Affine representation
• Character theory
• Great orthogonality theorem
• Maschke's theorem
• Monstrous moonshine
• Projective representation
• Representation theory
• Schur's lemma

[edit] Computational group theory

• Coset enumeration
• Schreier's subgroup lemma
• Schreier-Sims algorithm
• Todd-Coxeter algorithm

[edit] Applications

• Computer algebra system
• Cryptography
  • Discrete logarithm
  • Triple DES
  • Caesar cipher
• Exponentiating by squaring
• Knapsack problem
• Shor's algorithm
• Standard Model

[edit] Famous problems

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

[edit] 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
• Tarski monster group
• Thompson groups
• Transfer (group theory)

[edit] Major contributors to group theory

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 Ø. Ore

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