Engel group explained

In mathematics, an element x of a Lie group or a Lie algebra is called an n-Engel element,^[1] named after Friedrich Engel, if it satisfies the n-Engel condition that the repeated commutator [...[[''x'',''y''],y], ..., y]^[2]

Notes and References

1. Shumyatsky, P.. Tortora, A.. Tota, M.. An Engel condition for orderable groups. 21 Feb 2014. 1402.5247.
2. 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. gave examples of non-nilpotent Engel groups and algebras.

   Notes