In group theory, a branch of mathematics, 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 in 1934 and articulated by Wilhelm Magnus in 1937.[1] The process is sometimes called a "collection process".
The process can be generalized to define a totally ordered subset of a free non-associative algebra, that is, a free magma; this subset is called the Hall set. Members of the Hall set are binary trees; these can be placed in one-to-one correspondence with words, these being called the Hall words; the Lyndon words are a special case. Hall sets are used to construct a basis for a free Lie algebra, entirely analogously to the commutator collecting process. Hall words also provide a unique factorization of monoids.
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 = [''F''<sub>''n''</sub>, ''F''<sub>1</sub>]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
| n1 | |
| g=c | |
| 1 |
| n2 | |
| c | |
| 2 |
…
| nk | |
| c | |
| k |
c
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.