Engel group

In mathematics, an element x of a Lie group or a Lie algebra is called an n-Engel element,{{cite arXiv|author=Shumyatsky, P.|author2=Tortora, A.|author3=Tota, M.|title=An Engel condition for orderable groups|date=21 Feb 2014|arxiv=1402.5247}} named after Friedrich Engel, if it satisfies the n-Engel condition that the repeated commutator [...x,y],y], ..., y]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. with n copies of y is trivial (where [x, y] means xyx⁻¹y⁻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. Engel's theorem states that every finite-dimensional Engel algebra is nilpotent. {{harv|Cohn|1955}} gave examples of non-nilpotent Engel groups and algebras.