Engel group

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] 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. (Cohn 1955) gave examples of non-nilpotent Engel groups and algebras.

Notes

^ Shumyatsky, P.; Tortora, A.; Tota, M. (21 Feb 2014). "An Engel condition for orderable groups". arXiv:1402.5247.
^ 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.

Cohn, P. M. (1955), "A non-nilpotent Lie ring satisfying the Engel condition and a non-nilpotent Engel group", Proc. Cambridge Philos. Soc., 51 (3): 401–405, Bibcode:1955PCPS...51..401C, doi:10.1017/S0305004100030395, MR 0071720

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[1] Shumyatsky, P.; Tortora, A.; Tota, M. (21 Feb 2014). "An Engel condition for orderable groups". arXiv: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.

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