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[edit]

Basic properties of groups[edit]

Group homomorphisms[edit]

Basic types of groups[edit]

Simple groups and their classification[edit]

Classification of finite simple groups

Permutation and symmetry groups[edit]

Concepts groups share with other mathematics[edit]

Mathematical objects making use of a group operation[edit]

Mathematical fields and topics making important use of group theory[edit]

Algebraic structures related to groups[edit]

Group representations[edit]

Computational group theory[edit]

Applications[edit]

Famous problems[edit]

Group theorists[edit]