List Of Group Theory Topics

History, major contributors

• Joseph-Louis Lagrange
• Niels Abel
• Evariste Galois
• Augustin Louis Cauchy
• Arthur Cayley
• Otto Ludwig Hlder
• 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
• Fitting subgroup, generalized Fitting subgroup
• p-core

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
• Quasisimple 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
• Feit-Thompson theorem

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

• 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

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