A note on right-nil and strong-nil skew braces

TL;DR

This paper investigates properties of finite strong-nil skew braces, showing that they are not necessarily right-nilpotent unless certain conditions are met, thereby answering specific open questions in the field.

Contribution

It provides a complete answer to open questions about the nilpotency properties of strong-nil skew braces, clarifying conditions under which they are right-nilpotent.

Findings

Strong-nil skew braces of abelian type need not be right-nilpotent.

If a strong-nil skew brace is of nilpotent type and satisfies b * b = 0, then it is right-nilpotent.

Examples demonstrate the optimality of the conditions for right-nilpotency.

Abstract

The aim of this short note is to completely answer Questions 2.34 and 2.35 of arXiv:1806.01127. In particular, we show that a finite strong-nil skew brace $B$ of abelian type need not be right-nilpotent, but that this is the case if~$B$ is of nilpotent type and $b\ast b=0$ for all $b\in B$ (our examples show that this is the best possible result).