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 action Glossary of group theory List of group theory topics
Finite groups Classification of finite simple groups cyclic alternating Lie type sporadic Cauchy's theorem 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 (${\displaystyle \mathbb {Z} }$) Free group Modular groups PSL(2, ${\displaystyle \mathbb {Z} }$) SL(2, ${\displaystyle \mathbb {Z} }$) Arithmetic group Lattice Hyperbolic group
Topological and 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 Linear algebraic group Reductive group Abelian variety Elliptic curve
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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[]

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
• 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[]

Group theorists[]

See also[]

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