On Engel groups, nilpotent groups, rings, braces and the Yang-Baxter equation

It is shown that over an arbitrary field there exists a nil algebra R R whose adjoint group R o Rᵒ is not an Engel group. This answers a question by Amberg and Sysak from 1997. The case of an uncountable field also answers a recent question by Zelmanov. In 2007, Rump introduced braces and radical chains A n + 1 = A ⋅ A n Aⁿ⁺¹=A· Aⁿ and A ( n + 1 ) = A ( n ) ⋅ A A⁽ⁿ⁺¹⁾=A⁽ⁿ⁾· A of a brace A A . We show that the adjoint group A o Aᵒ of a finite right brace is a nilpotent group if and only if A ( n ) = 0 A⁽ⁿ⁾=0 for some n n . We also show that the adjoint group A o Aᵒ of a finite left brace A A is a nilpotent group if and only if A n = 0 Aⁿ=0 for some