# Non-abelian group

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In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group ( G, ∗ ) in which there exists at least one pair of elements a and b of G, such that a ∗ b ≠ b ∗ a. This class of groups contrasts with the abelian groups. (In an abelian group, all pairs of group elements 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 along one axis and then 90 degrees along a different axis is not the same as doing them the other way round).

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.