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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 . 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 ''F''1 is a free group on generators ''a''1, ..., ''a''''m''. Define the descending central series by putting :''F''''n''+1 = () The 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 () where ''x'' and ''y'' are basic commutators whose weights sum to ''w'', such that ''x'' > ''y'' and if ''x'' = () 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 fnitely-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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Commutator collecting process」の詳細全文を読む スポンサード リンク
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