Commutator collecting process


In mathematical group theory, the commutator collecting process is a method for writing an element of a group as a product of generators and their higher commutators arranged in a certain order. The commutator collecting process was introduced by. He called it a "collecting process" though it is also often called a "collection process".

Statement

The commutator collecting process is usually stated for free groups, as a similar theorem then holds for any group by writing it as a quotient of a free group.
Suppose F1 is a free group on generators a1, ..., am. Define the descending central series by putting
The basic commutators are elements of F1 defined and ordered as follows.
Commutators are ordered so that x > y if x has weight greater than that of y, and for commutators of any fixed weight some total ordering is chosen.
Then Fn/Fn+1 is a finitely generated free abelian group with a basis consisting of basic commutators of weight n.
Then any element of F can be written as
where the ci are the basic commutators of weight at most m arranged in order, and c is a product of commutators of weight greater than m, and the ni are integers.