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*.^{[1]}^{[2]} 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.

## See also

## References

**^**Dummit, David S.; Foote, Richard M. (2004).*Abstract Algebra*(3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.**^**Lang, Serge (2002).*Algebra*. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.