“Commutator collecting process”的意思、由来-开放百科全书

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Commutator collecting process

释义

Statement
References

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 {{harvs|txt|last=Hall|first=Philip|year=1934|authorlink=Philip Hall}}. 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, ..., a_m. Define the descending central series by putting

F_n+1 = [F_n, F₁]

The basic commutators are elements of F₁ defined and ordered as follows.

The basic commutators of weight 1 are the generators a₁, ..., a_m.
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 F_n/F_n+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 c_i 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 n_i are integers.

References

{{Citation | last1=Hall | first1=Marshall | author1-link=Marshall Hall (mathematician) | title=The theory of groups | publisher=Macmillan |mr=0103215 | year=1959}}
{{Citation | last1=Hall | first1=Philip | author1-link=Philip Hall | title=A contribution to the theory of groups of prime-power order | year=1934 | journal=Proceedings of the London Mathematical Society | volume=36 | pages=29–95 | doi=10.1112/plms/s2-36.1.29}}
{{Citation | last1=Huppert | first1=B. | author1-link=Bertram Huppert | title=Endliche Gruppen | publisher=Springer-Verlag | location=Berlin, New York | language=German | isbn=978-3-540-03825-2 | oclc=527050 |mr=0224703 | year=1967 | pages=90–93}}

2 : P-groups|Combinatorial group theory

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