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