List of group theory topics - Definition and Overview

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.

History, major contributors

• Joseph-Louis Lagrange
• Niels Abel
• Evariste Galois
• Augustin Louis Cauchy
• Arthur Cayley
• Otto Ludwig Hölder
• Camille Jordan
• Ludwig Sylow
• Ferdinand Georg Frobenius
• Sophus Lie
• Felix Klein
• William Burnside
• Richard Dedekind
• David Hilbert
• Max August Zorn
• John Thompson
• Martin Dunwoody
• John Conway

Famous problems

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

Applications

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

Definitions in common with other mathematical ideas

• Binary operation
• Bilinear operator
• Commutative
• Associativity
• Equivalence relation
• Equivalence class
• Bijection
• Multiplication table
• Lattice
• Up to
• Congruence relation
• Prime number

Mathematical objects which have (or make use of) a group operation

• Number
• Real number
• Integer
• Modular arithmetic
• Matrix
• Tensor
• Quaternion
  • Quaternion group
• Hilbert space
• Pauli matrices
• Gell-Mann matrices
• Banach-Tarski paradox
• Galois group
• Dimensional analysis
• Lie group
• Algebraic group
• Elliptic curve
• Abelian variety
• Group object
• Category of groups

Abstract algebra

• Quasigroup
• Semigroup
• Monoid
• Monoid ring
• Module
• Ring
• Finite field
• Field
• Vector space
• Linear algebra
• Galois theory

Other mathematical disciplines which make great use of groups

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

Basic properties of groups and subgroups

• Elementary group theory
• Identity element
• Reciprocal
• Subgroup
• Coset
• Transversal
• Lagrange's theorem
• Euler's theorem
• Normal subgroup
• Hamiltonian group
• Characteristic subgroup
• Torsion subgroup
• Centralizer and normalizer
• Center of a group
• Commutator
• Derived group
• Conjugate
• Conjugacy class
• Conjugate closure
• Core (group)
• Composition series
• Zassenhaus lemma, Butterfly lemma
• Schreier refinement theorem

Group homomorphisms

• Automorphism
• Automorphism group
• Group homomorphism
• Group isomorphism
• Factor group
• Quotient group
• Homomorphism
• Order automorphism
• Isomorphism theorem
• Fundamental theorem on homomorphisms
• Inner automorphism
• Outer automorphism group

Basic group structures and operations

• Generating set of a group
• Sylow theorem
• Product of groups
• Direct product
• Direct sum
• Direct sum of groups
• Subdirect sum
• Semidirect product
• Group Extension
• Central extension
• Group cohomology
• Wreath product
• Free product
• Free group
• Free abelian group
• Presentation of a group

Basic building block of groups, and classifications

• Examples of groups
• P-group
• Cyclic group
• Abelian group
  • Rank of an abelian group
• Finitely generated abelian group
• Nilpotent group
• Solvable group
• Dihedral group
• Dicyclic group
• Klein four-group
• Group representation
• Pro-finite group
• Locally cyclic group
• List of small groups
• Divisible group

Simple groups and their classification

• Simple group
• Symmetric group
• Alternating group
• General linear group
• Special linear group
• Projective group
• Mathieu group
• Janko group
• Conway group
• Fischer group
• Thompson group (finite)
• Tits group
• Baby Monster group
• Monster group
• Bimonster
• Lie group
• Simple Lie group
• Weyl group
• Algebraic group
  • Borel subgroup
  • Parabolic subgroup
  • Group scheme
• Chevalley group

Understanding groups as permutation and symmetry groups

• Rubik's Cube
• Symmetry
• Symmetry group
• Euclidean group
• Frieze group
• Wallpaper group
• Crystallographic group
• Crystallographic point group, Schoenflies notation
• Space group
• Coxeter group
• Discrete group
• Fuchsian group
• Arithmetic group
• Homogeneous space
• Permutation
• Permutation group
• Cayley's Theorem
• Even and odd permutations
• Strong generating set
• Braid group
• Group action
  • Orbit (group theory)
  • Stabilizer subgroup
• Burnside's lemma
• Steiner system
• Geometric group theory

Group representations

See also list of harmonic analysis and representation theory topics

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

Computational group theory

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

Other topics

• Frattini subgroup
• Transfer (group theory)
• Growth rate
• Heisenberg group, discrete Heisenberg group
• Congruence subgroup
• Commensurable
• Compactly generated group
• Complete group
• Thompson groups
• Tarski monster group
• Amenable group
• Compact group
• Covering a group by cosets or subgroups