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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. 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 nondegenerate involutive settheoretic solutions of the YangBaxter equation. It is worth noticing that by a result of GatevaIvanova braces are in onetoone correspondence with braided groups with involutive braiding operators.
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 nondegenerate involutive settheoretic solutions of the YangBaxter equation. It is worth noticing that by a result of GatevaIvanova braces are in onetoone correspondence with braided groups with involutive braiding operators.
Original language  English 

Pages (fromto)  65356564 
Number of pages  38 
Journal  Transactions of the American Mathematical Society 
Volume  370 
Issue number  9 
Early online date  20 Mar 2018 
DOIs  
Publication status  Published  Sep 2018 
Keywords
 Engel group, nilpotent group, adjoint group of a ring, braces, nil rings, nil algebras, the YangBaxter equation.
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Agata Smoktunowicz
 School of Mathematics  Personal Chair in Algebra
Person: Academic: Research Active