List of group theory topics

This order is by no means ideal; but it may help to provide some framework for a re-editing of the main topic. Nor are the entries below considered a definitive set of topics; rather they reflect the current state of group theory articles on Wikipedia. Feel free to add or subtract from this list as appropriate.

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 Category theory etc. 7 Abstract algebra 8 Other mathematical disciplines which make great use of groups 9 Basic properties of groups and subgroups 10 Group homomorphisms 11 Basic group structures and operations 12 Basic building block of groups, and classifications 12..1 Simple groups and their classification 13 Understanding groups as permutation groups

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

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
• Hilbert space
• Pauli matrices
• Banach-Tarski paradox
• Natural number
• Dimensional analysis
• Manifold
• Lie group
• Algebraic group
• Elliptic curve
• Abelian variety

• Morphism
• Topos
• Groupoid
• Limit (category theory)
• Preadditive category
• Initial object
• Yoneda lemma
• Natural transformation
• Embedding
• Standard Model

• Abstract algebra
• Universal algebra
• Algebra
• Subalgebra
• Quasigroup
• Semigroup
• Monoid
• Monoid ring
• Module
• Ring (algebra)
• Finite field
• Field
• Vector space
• Linear algebra
• Galois theory
• Galois group

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

• Rubik's Cube
• Symmetry
• Symmetry group
• Permutation
• Even permutation
• Permutation group
• Group action
• Cayley's Theorem
• Steiner system