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:

$$[a,b]=a^{-1}{b}^{-1}ab$$

• Right convention:

$$[a,b]=ab{a}^{-1}{b}^{-1}$$

These differ by inversion: $$ab{a}^{-1}{b}^{-1}=[b,a{]}^{-1}.$$

Unless otherwise stated, this article uses the left convention.

In ring theory and linear algebra, the standard convention is $$[A,B]=AB-BA.$$

Definition

Groups

Let $G$ be a group, and let $a,b\in G$. Their commutator is $$[a,b]=a^{-1}{b}^{-1}ab.$$

One has $$[a,b]=e⟺ab=ba,$$ where $e$ is the identity element.

Rings

Let $R$ be a ring, and let $A,B\in R$. Their commutator is $$[A,B]=AB-BA.$$

This vanishes precisely when $A$ and $B$ commute.

Linear transformations

For linear transformations $A,B:V\to V$ on a vector space $V$, equivalently for square matrices, the commutator is $$[A,B]=AB-BA.$$

This is the ring commutator in the endomorphism ring $\mathrm{End}(V)$.

Properties

Group identities

For all $a,b,c\in G$, $$[a,b{]}^{-1}=[b,a],$$

$$[a,a]=e,$$

$$[a,b{]}^c=[a^c,b^c],$$ where $$x^y=y^{-1}xy.$$

Also, $$[ab,c]=[a,c{]}^b[b,c],$$

$$[a,bc]=[a,c][a,b{]}^c.$$

Ring identities

For all $A,B,C$, $$[A,B]=-[B,A],$$

$$[A+B,C]=[A,C]+[B,C],$$

$$[A,BC]=[A,B]C+B[A,C].$$

The commutator also satisfies the Jacobi identity: $$[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0.$$

For matrices, $$\mathrm{tr}([A,B])=0.$$

Derived subgroup

The subgroup generated by all group commutators is the derived subgroup or commutator subgroup of $G$: $$G^{\prime }=[G,G]=⟨[a,b]\mid a,b\in G⟩.$$

It satisfies:

• $G^{\prime }$ is a normal subgroup
• $G/G^{\prime }$ is abelian
• $G$ is abelian if and only if $G^{\prime }=\{e\}$

The quotient $$G^{\mathrm{a}\mathrm{b}}=G/G^{\prime }$$ is called the abelianization of $G$.

Lie algebras

In a Lie algebra, the bracket operation often arises from the commutator in an associative algebra: $$[x,y]=xy-yx.$$

Thus every associative algebra determines a Lie algebra by using the commutator as its bracket.

For the matrix algebra $M_n(F)$, this gives the Lie algebra $$𝔤𝔩(n,F).$$

Examples

Symmetric group

In the symmetric group $S_3$, let $$a=(12),b=(23).$$

Then $$[a,b]=(132),$$ so $a$ and $b$ do not commute.

Matrices

Let $$A=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),B=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right).$$

Then $$[A,B]=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right).$$

Hence $A$ and $B$ do not commute.

See also

• Derived subgroup
• Lower central series
• Lie algebra
• Jacobi identity
• Abelianization
• Noncommutative ring