Categories
  Encyclosphere.org ENCYCLOREADER
  supported by EncyclosphereKSF

Non-abelian group

From HandWiki - Reading time: 3 min

Short description: Group where ab = ba does not always hold

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

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




Licensed under CC BY-SA 3.0 | Source: https://handwiki.org/wiki/Non-abelian_group
4 views | Status: cached on October 16 2024 15:25:32
↧ Download this article as ZWI file
Encyclosphere.org EncycloReader is supported by the EncyclosphereKSF