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

• for all [math]\displaystyle{ x, y }[/math] in [math]\displaystyle{ L }[/math], the n-Engel identity

[math]\displaystyle{ [x,[x, \ldots, [x,[x,y]]\ldots]] = 0 }[/math] (n copies of [math]\displaystyle{ x }[/math]), is satisfied.

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

See also

• Adjoint representation
• Efim Zelmanov
• Engel's theorem