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 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.
- The basic commutators of weight 1 are the generators a1, ..., am.
- The basic commutators of weight w > 1 are the elements [x, y] where x and y are basic commutators whose weights sum to w, such that x > y and if x = [u, v] for basic commutators u and v then y ≥ v.
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.
References
- Hall, Marshall (1959), The theory of groups, Macmillan, MR 0103215
- Hall, Philip (1934), "A contribution to the theory of groups of prime-power order", Proceedings of the London Mathematical Society, 36: 29–95, doi:10.1112/plms/s2-36.1.29
- Huppert, B. (1967), Endliche Gruppen (in German), Berlin, New York: Springer-Verlag, pp. 90–93, ISBN 978-3-540-03825-2, MR 0224703, OCLC 527050