Non-abelian group

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

Algebraic structure → Group theory
Group theory

Basic notions

* Subgroup
* Normal subgroup

* Quotient group
* (Semi-)direct product

Group homomorphisms

* kernel
* image
* direct sum

* wreath product
* simple
* finite

* infinite
* continuous
* multiplicative

* additive
* cyclic
* abelian
* dihedral

* nilpotent
* solvable

* List of group theory topics

Finite groups

|  Classification of finite simple groups

* cyclic
* alternating
* Lie type
* sporadic

* Lagrange's theorem

* Sylow theorems
* Hall's theorem

* p-group
* Elementary abelian group

* Frobenius group

* Schur multiplier

* Symmetric group Sn

* Klein four-group V
* Dihedral group Dn
* Quaternion group Q
* Dicyclic group Dicn

* Discrete groups
* Lattices

* Integers (Z)
* Free group

Modular groups

* PSL(2,Z)
* SL(2,Z)

* Arithmetic group
* Lattice
* Hyperbolic group

Topological and Lie groups

* Solenoid
* Circle

* General linear GL(n)

* Special linear SL(n)

* Orthogonal O(n)

* Euclidean E(n)

* Special orthogonal SO(n)

* Unitary U(n)

* Special unitary SU(n)

* Symplectic Sp(n)

* G2
* F4
* E6
* E7
* E8

* Lorentz
* Poincaré
* Conformal

* Diffeomorphism
* Loop

Infinite dimensional Lie group

* O(∞)
* SU(∞)
* Sp(∞)

Algebraic groups

* Linear algebraic group

* Reductive group

* Abelian variety

* Elliptic curve

* v
* t
* e

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

* Associative algebra
* Noncommutative geometry
* Niels Henrik Abel

## External links

* David Madore: Orders of non abelian simple groups. 22 January 2003. Retrieved 13 July 2024.

## 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.

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