List of group theory topics

See also: List of abstract algebra topics, List of category theory 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

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
• Banach-Tarski paradox
• Galois group
• Dimensional analysis
• Lie group
• Algebraic group
• Elliptic curve
• Abelian variety
• Group object

Abstract algebra

• Quasigroup
• Semigroup
• Monoid
• Monoid ring
• Module
• Ring (algebra)
• 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
• Theorem of Lagrange
• Euler's theorem
• Normal subgroup
• Hamiltonian group
• Characteristic subgroup
• Torsion subgroup
• Centralizer and normalizer
• Stabilizer subgroup
• Center of a group
• Commutator
• Derived group
• Conjugate
• Conjugacy class
• Conjugate closure
• Core (group)

Group homomorphisms

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

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
• Group cohomology
• Wreath product
• Free product
• Free group
• Free abelian group
• Presentation of a group

Basic building block of groups, and classifications

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

Simple groups and their classification

• Simple group
• Symmetric group
• Alternating group
• General linear group
• Special linear group
• Mathieu group
• Janko group
• Conway group
• Fischer group
• Baby Monster group
• Monster group
• Bimonster
• Lie group
• Simple Lie group
• Weyl group
• Algebraic group
• Chevalley group

Understanding groups as permutation and symmetry groups

• Rubik's Cube
• Symmetry
• Symmetry group
• Euclidean group
• Frieze group
• Wallpaper group
• Crystallographic group
• Homogeneous space
• Permutation
• Even permutation
• Permutation group
• Group action
• Cayley's Theorem
• Orbit (mathematics)
• Burnside's lemma
• Steiner system

Group representations

• Representation theory
• Affine representation
• Projective representation

Referenced By