|Algebraic structure → Group theory
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. 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. It is the smallest finite non-abelian 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 isn't the same as doing them the other way round), which is also called the quaternion group.