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The Engel identity, named after Friedrich Engel, is a mathematical equation that is satisfied by all elements of a Lie ring, in the case of an Engel Lie ring, or by all the elements of a group, in the case of an Engel group. The Engel identity is the defining condition of an Engel group.

Formal definition

A Lie ring ${\displaystyle L}$ is defined as a nonassociative ring with multiplication that is anticommutative and satisfies the Jacobi identity with respect to the Lie bracket ${\displaystyle }$, defined for all elements ${\displaystyle x,y}$ in the ring ${\displaystyle L}$. The Lie ring ${\displaystyle L}$ is defined to be an n-Engel Lie ring if and only if

• for all ${\displaystyle x,y}$ in ${\displaystyle L}$, the n-Engel identity

${\displaystyle ]\ldots ]]=0}$ (n copies of ${\displaystyle x}$), is satisfied.

In the case of a group ${\displaystyle G}$, in the preceding definition, use the definition = x • y • x • y and replace ${\displaystyle 0}$ by ${\displaystyle 1}$, where ${\displaystyle 1}$ is the identity element of the group ${\displaystyle G}$.

See also

• Adjoint representation
• Efim Zelmanov
• Engel's theorem

References

1. Traustason, Gunnar (1993). "Engel Lie-Algebras". Quart. J. Math. Oxford. 44 (3): 355–384. doi:10.1093/qmath/44.3.355.
2. Traustason, Gunnar. "Engel groups (a survey)" (PDF).

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