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 condition that it is 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 a non-nilpotent Engel groups and algebras.

References

*Citation | last1=Cohn | first1=P. M. | title=A non-nilpotent Lie ring satisfying the Engel condition and a non-nilpotent Engel group | id=MathSciNet | id = 0071720 | year=1955 | journal=Proc. Cambridge Philos. Soc. | volume=51 | pages=401–405