
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 [...[["x","y"],"y"], ..., "y"] with "n" copies of "y" is trivial (where ["x", "y"] means "xyx"−1"y"−1 or the [[Lie bracket]]). It is called an Engel element if it satisfies the [[Engel identity|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. gave examples of non-nilpotent Engel groups and algebras.
Notes.
[[Category:Group theory]]
[[Category:Lie algebras]]