In mathematics, a non-abelian group, also sometimes called a non-commutative group, is a group (G, * ) in which there are at least two elements a and b of G such that a * b ≠ b * a.[1][2] The term non-abelian is used to distinguish from the idea of an abelian group, where all of the elements of the group commute.
Non-abelian groups are pervasive in mathematics and physics. One of the simplest examples of a non-abelian group is the dihedral group of order 6. A common example from physics is the rotation group in three dimensions (rotating something 90 degrees away from you and the 90 degrees to the left isn't the same as doing them the other way round), which is also called the quaternion group.
Both discrete groups and continuous groups may be non-abelian. Most of the interesting Lie groups are non-abelian, and these play an important role in gauge theory.
| Concepts in group theory | ||||
| category of groups | ||||
| subgroups, normal subgroups | ||||
| group homomorphisms, kernel, image, quotient | ||||
| direct product, direct sum | ||||
| semidirect product, wreath product | ||||
| Types of groups | ||||
|---|---|---|---|---|
| simple, finite, infinite | ||||
| discrete, continuous | ||||
| multiplicative, additive | ||||
| cyclic, abelian, dihedral | ||||
| nilpotent, solvable | ||||
| list of group theory topics | ||||
| glossary of group theory | ||||