Non-abelian group

In mathematics, a nonabelian group, also sometimes called a noncommutative 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 nonabelian 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 nonabelian group is the dihedral group of order 6. It is the smallest finite nonabelian group. A common example from physics is the rotation group SO(3) in three dimensions (rotating something 90 degrees away from you and then 90 degrees to the left is not the same as doing them the other way round).

Both discrete groups and continuous groups may be nonabelian. Most of the interesting Lie groups are nonabelian, and these play an important role in gauge theory.

See also

• Abelian group
• Associative algebra
• Noncommutative geometry

References