Nonabelian group

In mathematics, a nonabelian group, also sometimes called a non-commutative group, is a group (G, * ) such that there are at least two elements a and b of G such that a * b ≠ b * a. The term non-abelian is used to distinguish from the idea of an abelian group, where all of the elements of the group commute.

Nonabelian groups are pervasive in mathematics and physics. One of the simplest examples of a non-abelian group is the symmetric group S₃. A common example from physics is the rotation group in three dimensions.

Both discrete groups and continuous groups may be non-abelian; most of the interesting Lie groups are nonabelian. The term nonabelian is used primarily by physicists, rather than mathematicians, and is frequently taken to be a synonym for the collection of Lie groups. This usage is particularly common in gauge theory.

See also

• Associative algebra
• Non-commutative geometry