Engel group
Encyclopedia
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−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 a non-nilpotent Engel groups and algebras.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an element x of a Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
or a Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
is called an n-Engel element, named after Friedrich Engel
Friedrich Engel (mathematician)
Friedrich Engel was a German mathematician.Engel was born in Lugau, Saxony, as the son of a Lutheran pastor. He attended the Universities of both Leipzig and Berlin, before receiving his doctorate from Leipzig in 1883.Engel studied under Felix Klein at Leipzig, and collaborated with Sophus Lie for...
, 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−1y−1 or the Lie bracket
Lie bracket
Lie bracket can refer to:*A bilinear binary operation defined on elements of a Lie algebra*Lie bracket of vector fields...
). 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
Nilpotent group
In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...
or Lie algebra is Engel. Engel's theorem states that every finite-dimensional Engel algebra is nilpotent. gave examples of a non-nilpotent Engel groups and algebras.