List of group theory topics

List of group theory topics
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.
Various physical systems, such as crystals and the hydrogen atom, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also central to public key cryptography.
Structures and operations
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
Parabolic subgroup
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 & 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
Congruence subgroup
Continuous symmetry
Frattini subgroup
Growth rate
Heisenberg group, discrete Heisenberg group
Molecular symmetry
Nielsen transformation
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
Mathematics-related lists
Outlines of mathematics and logic
Outlines