mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, an element ''x'' of a

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...

or a

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...

is called an ''n''-Engel element, named after Friedrich Engel, if it satisfies the ''n''-Engel condition that the repeated commutator ..''x'',''y''''y''.html" ;"title="''x'',''y''.html" ;"title="..''x'',''y''">..''x'',''y''''y''">''x'',''y''.html" ;"title="..''x'',''y''">..''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 ''x''⁻¹''y''⁻¹''xy'' or the

Lie bracket In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identit ...

). 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 In mathematics, specifically group theory, a nilpotent group ''G'' is a group that has an upper central series that terminates with ''G''. Equivalently, its central series is of finite length or its lower central series terminates with . Intuiti ...

or Lie algebra is Engel.

Engel's theorem In representation theory, a branch of mathematics, Engel's theorem states that a finite-dimensional Lie algebra \mathfrak g is a nilpotent Lie algebra_if_and_only_if_for_each_X_\in_\mathfrak_g,_the_adjoint_representation_of_a_Lie_algebra.html" "ti ...

states that every finite-dimensional Engel algebra is nilpotent. gave examples of non-nilpotent Engel groups and algebras.

Notes

*{{Citation , last1=Cohn , first1=P. M. , authorlink = Paul Cohn, title=A non-nilpotent Lie ring satisfying the Engel condition and a non-nilpotent Engel group , mr=0071720 , year=1955 , journal=Proc. Cambridge Philos. Soc. , volume=51 , pages=401–405 , doi=10.1017/S0305004100030395 , issue=3, bibcode=1955PCPS...51..401C Group theory Lie algebras