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

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Original languageEnglish
Pages (from-to)6535-6564
Number of pages38
JournalTransactions of the American Mathematical Society
Issue number9
Early online date20 Mar 2018
Publication statusPublished - Sep 2018


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. 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\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 $ 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$ is a finite brace whose adjoint group $ A^{o}$ is nilpotent, then $ A$ is the direct sum of braces whose cardinalities are powers of prime numbers. Notice that $ A^{o}$ is sometimes called the multiplicative group of a brace $ A$. 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 equation. It is worth noticing that by a result of Gateva-Ivanova braces are in one-to-one correspondence with braided groups with involutive braiding operators.

    Research areas

  • Engel group, nilpotent group, adjoint group of a ring, braces, nil rings, nil algebras, the Yang-Baxter equation.

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