Commutator collecting process

In mathematical group theory, the commutator collect-ing process is a method for writing an element of a groupas a product of generators and their higher commutatorsarranged in a certain order. The commutator collectingprocess was introduced by Philip Hall (1934). He calledit a “collecting process” though it is also often called a“collection process”.

1 Statement

The commutator collecting process is usually stated forfree groups, as a similar theorem then holds for any groupby writing it as a quotient of a free group.Suppose F1 is a free group on generators a1, ..., am. De-fine the descending central series by putting

Fn₊₁ = [Fn, F1]

The basic commutators are elements of F1 defined andordered as follows.

• The basic commutators of weight 1 are the genera-tors a1, ..., am.

• The basic commutators of weight w > 1 are the el-ements [x, y] where x and y are basic commutatorswhose 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 weightgreater than that of y, and for commutators of any fixedweight some total ordering is chosen.Then Fn/Fn₊₁ is a fnitely-generated free abelian groupwith a basis consisting of basic commutators of weightn.Then any element of F can be written as

g = cn11 cn2

2 · · · cnk

k c

where the ci are the basic commutators of weight at mostm arranged in order, and c is a product of commutatorsof weight greater than m, and the ni are integers.

2 References• Hall, Marshall (1959), The theory of groups,Macmillan, MR 0103215

• Hall, Philip (1934), “A contribution to the the-ory of groups of prime-power order”, Proceedingsof 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, OCLC527050

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2 3 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

3 Text and image sources, contributors, and licenses

3.1 Text• Commutator collecting process Source: https://en.wikipedia.org/wiki/Commutator_collecting_process?oldid=653635876 Contributors:

Michael Hardy, R.e.b. and K9re11

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