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

Table of contents

1 History, major contributors 2 Famous problems 3 Applications 4 Definitions in common with other mathematical ideas 5 Mathematical objects which have (or make use of) a group operation 6 Abstract algebra 7 Other mathematical disciplines which make great use of groups 8 Basic properties of groups and subgroups 9 Group homomorphisms 10 Basic group structures and operations 11 Basic building block of groups, and classifications 11..1 Simple groups and their classification 12 Understanding groups as permutation and symmetry groups 13 Group representations

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

• Classification of finite simple groups
• Word problem for groups
• Subset sum problem
• Burnside's problem
• Whitehead problem

• 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

• 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

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

• 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)

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

• 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

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

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

• Representation theory
• Affine representation
• Projective representation