List of group theory topics

Algebraic structure → Group theory Group theory
Basic notions Subgroup Normal subgroup Quotient group (Semi-)direct product Group homomorphisms kernel image direct sum wreath product simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable List of group theory topics Glossary of group theory
Finite groups Classification of finite simple groups cyclic alternating Lie type sporadic Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group Schur multiplier Symmetric group Sn Klein four-group V Dihedral group Dn Quaternion group Q Dicyclic group Dicn
Discrete groups Lattices Integers (Z) Lattice Modular groups PSL(2,Z) SL(2,Z)
Topological / Lie groups Solenoid Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞)
Algebraic groups Elliptic curve Linear algebraic group Abelian variety

Structures and operations

Basic properties of groups

Group homomorphisms

Basic types of groups

Simple groups and their classification

• Classification of finite simple groups

Permutation and symmetry groups

Concepts groups share with other mathematics

Mathematical objects making use of a group operation

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

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

Famous problems

• Burnside's problem
• Classification of finite simple groups
• Herzog-Schönheim conjecture
• Subset sum problem
• Whitehead problem
• Word problem for groups

Other topics

Group theorists

See also

• List of abstract algebra topics
• List of category theory topics
• List of Lie group topics