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 Philip Hall (1934). 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
Fn+1 = [Fn, F1]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.