In mathematics, an element ''x'' of a Lie group or a Lie algebra is called an '''''n''-Engel element''',<ref>{{cite arXiv|author=Shumyatsky, P.|author2=Tortora, A.|author3=Tota, M.|title=An Engel condition for orderable groups|date=21 Feb 2014|class=math.GR |eprint=1402.5247}}</ref> named after Friedrich Engel, if it satisfies the '''''n''-Engel condition''' that the repeated commutator [...[[''x'',''y''],''y''], ..., ''y'']<ref>In other words, ''n'' "["s and n copies of y, for example, [[[x,y],y],y], [[[[x,y],y],y],y]. [[[[[x,y],y],y],y],y], and so on.</ref> with ''n'' copies of ''y'' is trivial (where [''x'', ''y''] means ''xyx''<sup>−1</sup>''y''<sup>−1</sup> 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. Engel's theorem states that every finite-dimensional Engel algebra is nilpotent. {{harv|Cohn|1955}} gave examples of non-nilpotent Engel groups and algebras.
==Notes== {{reflist}} *{{Citation | last1=Cohn | first1=P. M. |authorlink = Paul Cohn| title=A non-nilpotent Lie ring satisfying the Engel condition and a non-nilpotent Engel group | mr=0071720 | year=1955 | journal=Proc. Cambridge Philos. Soc. | volume=51 | pages=401–405 | doi=10.1017/S0305004100030395 | issue=3| bibcode=1955PCPS...51..401C }}
Category:Group theory Category:Lie algebras
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