List of group theory topics

This page collects all articles on group theory, 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.

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.

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History of; major contributors to

• 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 in

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

Applications of

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

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

Category theory and

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

Abstract Algebra and

• 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

Other mathematical disciplines which make great use of groups

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

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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
• 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

• Simple group
• P-group
• Cyclic group
• Abelian group
• Nilpotent group
• Solvable group
• Dihedral group
• Klein four-group
• Symmetric group
• Alternating group
• General linear group
• Special linear group
• Group representation
• Pro-finite group
• Monster group
• Lie group
• List of small groups
• Finitely generated abelian group
• Divisible group[?]

Understanding groups as permutation groups

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