Commutator
A commutator is an algebraic expression that measures the failure of two elements to commute. It occurs throughout abstract algebra, particularly in group theory, ring theory, and linear algebra.
If two elements commute, their commutator is trivial. More generally, the commutator describes the obstruction to exchanging the order of two operations. Commutators are fundamental in the study of noncommutative structures and in the construction of invariants such as the derived subgroup, the lower central series, and the Lie bracket.
Conventions
In group theory, two conventions are commonly used:
- Left convention:
- Right convention:
These differ by inversion:
Unless otherwise stated, this article uses the left convention.
In ring theory and linear algebra, the standard convention is
Definition
Groups
Let 
be a group, and let 
. Their commutator is
One has
where 
is the identity element.
Rings
Let 
be a ring, and let 
. Their commutator is
This vanishes precisely when 
and 
commute.
Linear transformations
For linear transformations 
on a vector space 
, equivalently for square matrices, the commutator is
This is the ring commutator in the endomorphism ring 
.
Properties
Group identities
For all 
,
where
Also,
Ring identities
For all 
,
The commutator also satisfies the Jacobi identity:
For matrices,
Derived subgroup
The subgroup generated by all group commutators is the derived subgroup or commutator subgroup of :
It satisfies:
is a normal subgroup
is abelian- is abelian if and only if
The quotient
is called the abelianization of .
Lie algebras
In a Lie algebra, the bracket operation often arises from the commutator in an associative algebra:
Thus every associative algebra determines a Lie algebra by using the commutator as its bracket.
For the matrix algebra 
, this gives the Lie algebra
Examples
Symmetric group
In the symmetric group 
, let
Then
so 
and 
do not commute.
Matrices
Let
Then
Hence and do not commute.