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.
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.
Hall sets are used to construct a basis for a free Lie algebra, entirely analogously to the commutator collecting process.
Then Fn /Fn+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 ci 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 ni are integers.