Engel identity

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

for all $x, y$ in $L$ , the n-Engel identity

$[x,[x, \ldots, [x,[x,y]],\ldots]] = 0$ (n copies of $x$ ), is satisfied.^[1]

In the case of a group $G$ , in the preceding definition, use the definition [x,y] = x⁻¹ • y⁻¹ • x • y and replace $0$ by $1$ , where $1$ is the identity element of the group $G$ .^[2]

References

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

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Engel identity

Formal definition

See also

References