Definition with symbols
The term commutator is used in group theory in two senses: the commutator map (typically with respect to the left action convention), and the set of possible images of that map. The commutator map is the following map:
And is denoted by outside square brackets (that is, the commutator of and is denoted as ). The image of this map is termed the commutator of and .
An element of the group is termed a commutator if it occurs as the commutator of some two elements of the group.
Sometimes the commutator map is defined somewhat differently (typically with respect to the right action convention), i.e., as:
Although this alters the commutator map, it does not alter the set of elements that are commutators, because the commutator of and in one definition becomes the commutator of and in the other definition.
Identity element is a commutator
In fact, whenever , . Hence, in particular, the commutator of any element with itself is the identity element.
Inverse of a commutator is a commutator
The inverse of the commutator is the commutator (this statement is true regardless of which definition of commutator we follow).
Product of commutators need not be a commutator
In general, a product of commutators need not be a commutator.
Since the commutator of two elements of the group is itself an element of the group, it can be treated as one of the inputs to the commutator map yet again. This allows us to construct iterated commutators.
A simple commutator is an iterated commutator that can be expressed as a left-normed commutator, i.e., a commutator of the form:
The subgroup generated by all the commutators is termed the derived subgroup.