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

Abstract

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.

To appear in the Transactions of the AMS. Improved the presentation, corrected a few typos

References in corpus (5)

• Extensions of set-theoretic solutions of the Yang-Baxter equation and a conjecture of Gateva-Ivanova
• Notes on Engel groups and Engel elements in groups. Some generalizations
• Nilpotent symplectic alternating algebras I
• How far can we go with Amitsur's theorem?
• Iterated matched products of finite braces and simplicity; new solutions of the Yang-Baxter equation

Cited by in corpus (3)

• Hopf braces and Yang-Baxter operators
• A note on set-theoretic solutions of the Yang-Baxter equation
• Towards semi-trusses