# Non-abelian group

__: Group where ab = ba does not always hold__

**Short description**Algebraic structure → Group theoryGroup theory |
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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*.^{[1]}^{[2]} This class of groups contrasts with the abelian groups, where 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 (for example, rotating something 90 degrees along one axis and then 90 degrees along a different axis is not the same as doing them in reverse order).

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.

Original source: https://en.wikipedia.org/wiki/Non-abelian group.
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