It is shown that over an arbitrary field there exists a nil algebra
R whose adjoint group is not an Engel group. This answers a question by
Amberg and Sysak from 1997. The case of an uncountable eld also answers
a recent question by Zelmanov.
In 2007, Rump introduced braces and radical chains A^{n+1} = A * A^n and
A^(n+1) = A^(n)* A of a brace A. We show that the adjoint group of a
finite right brace is a nilpotent group if and only if A^(n) = 0 for some n. We
also show that the adjoint group of a finite left brace A is a nilpotent
group if and only if A^n = 0 for some n. Moreover, if A is a finite brace
whose adjoint group is nilpotent then A is the direct sum of braces whose
cardinatities are powers of prime numbers. We also introduce a chain of ideals A^[n]
of a left brace A and then use it to investigate braces which satisfy A^n = 0
and A^(m) = 0 for some m; n.
We also describe connections between our results and braided groups and
the non-degenerate involutive set-theoretic solutions of the Yang-Baxter equa-
tion. It is worth noticing that by a result by Gateva-Ivanova braces are in
one-to-one correspondence with braided groups with involutive braiding op-
erators.