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

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 structures related to groups

Group representations

See also: List of representation theory topics and List of harmonic analysis topics

Computational group theory

Main article: Computational group theory


Famous problems

Other topics

Group theorists

See also