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 R whose adjoint group R o 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 ⋅ A n A^{n+1}=A\cdot A^{n} and A ( n + 1 ) = A ( n ) ⋅ A A^{(n+1)}=A^{(n)}\cdot A of a brace A A . We show that the adjoint group A o A^{o} of a finite right brace is a nilpotent group if and only if A ( n ) = 0 A^{(n)}=0 for some n n . We also show that the adjoint group A o A^{o} of a finite left brace A A is a nilpotent group if and only if A n = 0 A^{n}=0 for some n n . Moreover, if A A is a finite brace whose adjoint group A o A^{o} is nilpotent, then A A is the direct sum of braces whose cardinalities are powers of prime numbers. Notice that A o A^{o} is sometimes called the multiplicative group of a brace A A . We also introduce a chain of ideals A [ n ] A^{[n]} of a left brace A A and then use it to investigate braces which satisfy A n = 0 A^{n}=0 and A ( m ) = 0 A^{(m)}=0 for some m , n 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.