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On left nilpotent skew braces of class
$2$
- Part of
-
- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 20 July 2026, pp. 1-7
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- Article
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The main objective of this article is to present a detailed structure of left nilpotent skew braces
$B$ of class 
$2$, i.e. skew braces with 
$B^3 = 0$. We prove that if 
$B$ is of nilpotent type, then 
$B$ is centrally nilpotent. In fact, we show that 
$B$ is right nilpotent of class at most 
$2+mr$, i.e. 
$B^{(2+mr+1)} = 0$, where 
$m$ and 
$r$ are the nilpotency classes of the additive group of 
$B$ and 
$B^2$, respectively. If 
$B$ is of abelian type, then 
$B$ is actually right nilpotent of class 
$3$, i.e. 
$B^{(4)} = 0$, and this bound is the best possible.
