In 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, and three of the four known fundamental forces in the universe, 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 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
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