Nonabelian group
Encyclopedia
In mathematics
, a non-abelian group, also sometimes called a non-commutative group, is a group
(G, * ) in which there are at least two elements a and b of G such that a * b ≠ b * a. The term non-abelian is used to distinguish from the idea of an abelian group
, where all of the elements of the group 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
. A common example from physics is the rotation group
in three dimensions (rotating something 90 degrees away from you and the 90 degrees to the left isn't the same as doing them the other way round), which is also called the quaternion group
.
Both discrete group
s and continuous groups may be non-abelian. Most of the interesting Lie group
s are non-abelian, and these play an important role in gauge theory
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a non-abelian group, also sometimes called a non-commutative group, is a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
(G, * ) in which there are at least two elements a and b of G such that a * b ≠ b * a. The term non-abelian is used to distinguish from the idea of an abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
, where all of the elements of the group commute
Commutativity
In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...
.
Non-abelian groups are pervasive in mathematics and physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
. One of the simplest examples of a non-abelian group is the dihedral group of order 6
Dihedral group of order 6
The smallest non-abelian group has 6 elements. It is a dihedral group with notation D3 and the symmetric group of degree 3, with notation S3....
. A common example from physics is the rotation group
Rotation group
In mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition. By definition, a rotation about the origin is a linear transformation that preserves length of vectors and preserves orientation ...
in three dimensions (rotating something 90 degrees away from you and the 90 degrees to the left isn't the same as doing them the other way round), which is also called the quaternion group
Quaternion group
In group theory, the quaternion group is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication...
.
Both discrete group
Discrete group
In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...
s and continuous groups may be non-abelian. Most of the interesting Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s are non-abelian, and these play an important role in gauge theory
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
.