Commutator

Definition

Definition with symbols

The term commutator is used in group theory in two senses: the commutator map, and the set of possible images of that map. The commutator map is the following map:

$(x,y)\mapsto x^{-1}y^{-1}xy$

And is denoted by outside square brackets (that is, the commutator of $x$ and $y$ is denoted as $[x,y]$). The image of this map is termed the commutator of $x$ and $y$.

An element of the group is termed a commutator if it occurs as the commutator of some two elements of the group.

Facts

Identity element is a commutator

In fact, whenever $xy=yx$, $[x,y]=e$. Hence, in particular, the commutator of any element with itself is a commutator.

Inverse of a commutator is a commutator

The inverse of the commutator $[x,y]$ is the commutator $[y,x]$.

Product of commutators need not be a commutator

In general, a product of commutators need not be a commutator.

Iterated commutators

Since the commutator of two elements of the group is itself a commutator, it can be treated as one of the inputs to the commutator map yet again. This allows us to construct iterated commutators.

A basic commutator is a word that can be expressed purely by iterating the commutator operations.

Subgroups related to the commutator

The subgroup generated by all the commutators is termed the commutator subgroup.