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In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. == Group theory == The commutator of two elements, ''g'' and ''h'', of a group ''G'', is the element :(''h'' ) = ''g''−1''h''−1''gh''. It is equal to the group's identity if and only if ''g'' and ''h'' commute (i.e., if and only if ''gh'' = ''hg''). The subgroup of ''G'' generated by all commutators is called the ''derived group'' or the ''commutator subgroup'' of ''G''. Note that one must consider the subgroup generated by the set of commutators because in general the set of commutators is not closed under the group operation. Commutators are used to define nilpotent and solvable groups. The above definition of the commutator is used by some group theorists, as well as throughout this article. However, many other group theorists define the commutator as :(''h'' ) = ''ghg''−1''h''−1. 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」の詳細全文を読む スポンサード リンク
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