Basic Commutator

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.
W. Magnus (1937), "Über Beziehungen zwischen höheren Kommutatoren", ''J. Grelle'' 177, 105-115.
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.

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 ''F''₁ is a free group on generators ''a''₁, ..., ''a''_''m''. Define the descending central series by putting
:''F''_''n''+1 =  'F''_''n'', ''F''₁The basic commutators are elements of ''F''₁ defined and ordered as follows:
*The basic commutators of weight 1 are the generators ''a''₁, ..., ''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 ''v'' ≤ ''y''.
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

:$g=c_1^c_2^\cdots c_k^c$

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.

See also

* Hall–Petresco identity
* Monoid factorisation

References

Reading

*
*{{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

P-groups
Combinatorial group theory