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Engel group
In mathematics, an element ''x'' of a Lie group or a Lie algebra is called an ''n''-Engel element, named after Friedrich Engel, if it satisfies the ''n''-Engel condition that the repeated commutator ,''y''], ..., ''y'']〔In other words, ''n'' ",y], x,y],y],y],y]. (),y],y],y],y], and so on.〕 with ''n'' copies of ''y'' is trivial (where () means ''xyx''−1''y''−1 or the Lie bracket" TITLE="(),y],y], x,y],y],y],y]. (),y],y],y],y], and so on.〕 with ''n'' copies of ''y'' is trivial (where () means ''xyx''−1''y''−1 or the Lie bracket">(),y],y], x,y],y],y],y]. (),y],y],y],y], and so on.〕 with ''n'' copies of ''y'' is trivial (where () means ''xyx''−1''y''−1 or the Lie bracket). It is called an Engel element if it satisfies the Engel condition that it is ''n''-Engel for some ''n''.
A Lie group or Lie algebra is said to satisfy the Engel or ''n''-Engel conditions if every element does. Such groups or algebras are called Engel groups, ''n''-Engel groups, Engel algebras, and ''n''-Engel algebras.
Every nilpotent group or Lie algebra is Engel. (theorem )] states that every finite-dimensional Engel algebra is nilpotent. gave examples of a non-nilpotent Engel groups and algebras.
==Notes==
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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「Engel group」の詳細全文を読む
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