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In group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute. It is also true that finite nilpotent groups are supersolvable. Nilpotent groups arise in Galois theory, as well as in the classification of groups. They also appear prominently in the classification of Lie groups. Analogous terms are used for Lie algebras (using the Lie bracket) including nilpotent, lower central series, and upper central series. ==Definition== The definition uses the idea, explained on its own page, of a central series for a group. The following are equivalent formulations: * A nilpotent group is one that has a central series of finite length. * A nilpotent group is one whose lower central series terminates in the trivial subgroup after finitely many steps. * A nilpotent group is one whose upper central series terminates in the whole group after finitely many steps. For a nilpotent group, the smallest ''n'' such that ''G'' has a central series of length ''n'' is called the nilpotency class of ''G'' ; and ''G'' is said to be nilpotent of class ''n''. (By definition, the length is ''n'' if there are ''n'' + 1 different subgroups in the series, including the trivial subgroup and the whole group.) Equivalently, the nilpotency class of ''G'' equals the length of the lower central series or upper central series. If a group has nilpotency class at most ''m'', then it is sometimes called a nil-''m'' group. It follows immediately from any of the above forms of the definition of nilpotency, that the trivial group is the unique group of nilpotency class 0, and groups of nilpotency class 1 are exactly the non-trivial abelian groups.〔 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「Nilpotent group」の詳細全文を読む スポンサード リンク
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