Commutator collecting process

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^[1] and articulated by Wilhelm Magnus in 1937.^[2] 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 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 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

${\displaystyle g=c_{1}^{n_{1}}c_{2}^{n_{2}}\cdots c_{k}^{n_{k}}c}$

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