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Structures and operations
- Conjugacy class sum
- Central extension
- Direct product of groups
- Direct sum of groups
- Extension problem
- Free abelian group
- Free group
- Free product
- Generating set of a group
- Group cohomology
- Group extension
- Presentation of a group
- Product of group subsets
- Schur multiplier
- Semidirect product
- Sylow theorems
- Hall subgroup
- Wreath product
Basic properties of groups
- Butterfly lemma
- Center of a group
- Centralizer and normalizer
- Characteristic subgroup
- Commutator
- Composition series
- Conjugacy class
- Conjugate closure
- Conjugation of isometries in Euclidean space
- Core (group)
- Coset
- Derived group
- Euler's theorem
- Fitting subgroup
- Generalized Fitting subgroup
- Hamiltonian group
- Identity element
- Lagrange's theorem
- Multiplicative inverse
- Normal subgroup
- Perfect group
- p-core
- Schreier refinement theorem
- Subgroup
- Transversal (combinatorics)
- Torsion subgroup
- Zassenhaus lemma
Group homomorphisms
- Automorphism
- Automorphism group
- Factor group
- Fundamental theorem on homomorphisms
- Group homomorphism
- Group isomorphism
- Homomorphism
- Isomorphism theorem
- Inner automorphism
- Order automorphism
- Outer automorphism group
- Quotient group
Basic types of groups
- Examples of groups
- Abelian group
- Cyclic group
- Rank of an abelian group
- Dicyclic group
- Dihedral group
- Divisible group
- Finitely generated abelian group
- Group representation
- Klein four-group
- List of small groups
- Locally cyclic group
- Nilpotent group
- Non-abelian group
- Solvable group
- P-group
- Pro-finite group
Simple groups and their classification
- Classification of finite simple groups
- Alternating group
- Borel subgroup
- Chevalley group
- Conway group
- Feit–Thompson theorem
- Fischer group
- General linear group
- Group of Lie type
- Group scheme
- HN group
- Janko group
- Lie group
- Simple Lie group
- Linear algebraic group
- List of finite simple groups
- Mathieu group
- Monster group
- Baby Monster group
- Bimonster
- Projective group
- Reductive group
- Simple group
- Quasisimple group
- Special linear group
- Symmetric group
- Thompson group (finite)
- Tits group
- Weyl group
Permutation and symmetry groups
- Arithmetic group
- Braid group
- Burnside's lemma
- Cayley's theorem
- Coxeter group
- Crystallographic group
- Crystallographic point group, Schoenflies notation
- Discrete group
- Euclidean group
- Even and odd permutations
- Frieze group
- Frobenius group
- Fuchsian group
- Geometric group theory
- Group action
- Homogeneous space
- Hyperbolic group
- Isometry group
- Orbit (group theory)
- Permutation
- Permutation group
- Rubik's Cube group
- Space group
- Stabilizer subgroup
- Steiner system
- Strong generating set
- Symmetry
- Symmetric group
- Symmetry group
- Wallpaper group
Concepts groups share with other mathematics
- Associativity
- Bijection
- Bilinear operator
- Binary operation
- Commutative
- Congruence relation
- Equivalence class
- Equivalence relation
- Lattice (group)
- Lattice (discrete subgroup)
- Multiplication table
- Prime number
- Up to
Mathematical objects making use of a group operation
- Abelian variety
- Algebraic group
- Banach–Tarski paradox
- Category of groups
- Dimensional analysis
- Elliptic curve
- Galois group
- Gell-Mann matrices
- Group object
- Hilbert space
- Integer
- Lie group
- Matrix
- Modular arithmetic
- Number
- Pauli matrices
- Real number
- Quaternion
- Quaternion group
- Tensor
Mathematical fields and topics making important use of group theory
- Algebraic geometry
- Algebraic topology
- Discrete space
- Fundamental group
- Geometry
- Homology
- Minkowski's theorem
- Topological group
Algebraic structures related to groups
- Field
- Finite field
- Galois theory
- Grothendieck group
- Group ring
- Group with operators
- Heap
- Linear algebra
- Magma
- Module
- Monoid
- Monoid ring
- Quandle
- Quasigroup
- Quantum group
- Ring
- Semigroup
- Vector space
Group representations
- Affine representation
- Character theory
- Great orthogonality theorem
- Maschke's theorem
- Monstrous moonshine
- Projective representation
- Representation theory
- Schur's lemma
Computational group theory
- Coset enumeration
- Schreier's subgroup lemma
- Schreier–Sims algorithm
- Todd–Coxeter algorithm
Applications
- Computer algebra system
- Cryptography
- Discrete logarithm
- Triple DES
- Caesar cipher
- Exponentiating by squaring
- Knapsack problem
- Shor's algorithm
- Standard Model
- Symmetry in physics
Famous problems
- Burnside's problem
- Classification of finite simple groups
- Herzog–Schönheim conjecture
- Subset sum problem
- Whitehead problem
- Word problem for groups
Other topics
- Amenable group
- Capable group
- Commensurability (group theory)
- Compact group
- Compactly generated group
- Complete group
- Complex reflection group
- Congruence subgroup
- Continuous symmetry
- Frattini subgroup
- Growth rate
- Heisenberg group, discrete Heisenberg group
- Molecular symmetry
- Nielsen transformation
- Reflection group
- Tarski monster group
- Thompson groups
- Tietze transformation
- Transfer (group theory)
Group theorists
- N. Abel
- M. Aschbacher
- R. Baer
- R. Brauer
- W. Burnside
- R. Carter
- A. Cauchy
- A. Cayley
- J.H. Conway
- R. Dedekind
- L.E. Dickson
- M. Dunwoody
- W. Feit
- B. Fischer
- H. Fitting
- G. Frattini
- G. Frobenius
- E. Galois
- G. Glauberman
- D. Gorenstein
- R.L. Griess
- M. Hall, Jr.
- P. Hall
- G. Higman
- D. Hilbert
- O. Hölder
- B. Huppert
- K. Iwasawa
- Z. Janko
- C. Jordan
- F. Klein
- A. Kurosh
- J.L. Lagrange
- C. Leedham-Green
- F.W. Levi
- Sophus Lie
- W. Magnus
- E. Mathieu
- G.A. Miller
- B.H. Neumann
- H. Neumann
- J. Nielson
- Emmy Noether
- Ø. Ore
- O. Schreier
- I. Schur
- R. Steinberg
- M. Suzuki
- L. Sylow
- J. Thompson
- J. Tits
- Helmut Wielandt
- H. Zassenhaus
- M. Zorn
See also
- List of abstract algebra topics
- List of category theory topics
- List of Lie group topics