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 $R$ whose adjoint group $R^{o}$ is not an Engel group. This answers a question by Amberg and Sysak from 1997 [5] and answers related questions from [3, 44]. The case of an uncountable field also answers a recent question by Zelmanov. In [38], Rump introduced braces and radical chains $A^{n+1}=A\cdot A^{n}$ and $A^{(n+1)}=A^{(n)}\cdot A$ of a brace $A$. We show that the adjoint group $A^{o}$ 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 $A^{o}$ of a finite left brace $A$ is a nilpotent group if and only if $A^{n}=0$ for some $n$. Moreover, if $A^{o}$ is a nilpotent group then $A$ is the direct sum of braces whose cardinatities are powers of prime numbers. Notice that $A^{o}$ is sometimes called the multiplicative group of a brace $A$ (for example in [13]). 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 $m, n$ (Theorems 2, 3). In Section 2 we describe connections between our results and braided groups and the Yang-Baxter equation. It is worth noticing that by a result by Gateva-Ivanova [17] braces are in one-to-one correspondence with braided groups with involutive braided operators.