Commutator collecting process

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Commutator collecting process (simplified)

In group theory, the commutator collecting process is a way to write any element of a group as a product of basic commutators in a fixed order. It was introduced by Philip Hall in 1934 and later explained by Wilhelm Magnus in 1937. It is sometimes called a collection process.

How it works
- Start with a free group on generators a1, a2, ..., am.
- Build a descending central series F1, F2, ... where Fn+1 is the subgroup generated by all commutators [x,y] with x in Fn and y in F1.
- The basic commutators are defined by weight. Weight 1 pieces are the generators themselves. For weight w > 1, the basic commutators are of the form [x,y], where x and y are basic commutators whose weights add to w, with x > y in a fixed order, and if x = [u,v], then v ≤ y.
- All basic commutators are ordered by weight, and within the same weight by a chosen total order.
- Then Fn / Fn+1 is a free abelian group with a basis consisting of the basic commutators of weight n.

Expressing elements
- Any element g of the free group F can be written as a product g = c1^n1 c2^n2 ... ck^nk c, where each ci is a basic commutator of weight at most m, the ni are integers, and c is a product of commutators of weight greater than m. The Ci are taken in the fixed order.

Relation to Hall sets and Lyndon words
- The basic commutators correspond to a Hall set, and, as words, relate to Hall words; Lyndon words are a special case. Hall sets and words help build a basis for the free Lie algebra and give a unique factorization of monoids.

See also and references
- Hall–Petresco identity; Monoid factorisation.
- References: Hall (1934); Magnus (1937); Hall (1959); Huppert (1967).