# Retractability of solutions to the Yang-Baxter equation and $p$-nilpotency of skew braces

## TL;DR

This paper investigates the retractability of solutions to the Yang-Baxter equation using Bieberbach groups and introduces the theory of right and left p-nilpotent skew braces, extending previous results to non-involutive solutions.

## Contribution

It develops the theory of right and left p-nilpotent skew braces and applies Bieberbach groups to analyze the structure and retractability of solutions to the Yang-Baxter equation.

## Key findings

- Algorithm for identifying Promislow subgroups in Bieberbach groups
- Extension of retractability results to non-involutive solutions
- Short proof of Smoktunowicz's theorem using skew braces

## Abstract

Using Bieberbach groups we study multipermutation involutive solutions to the Yang-Baxter equation. We use a linear representation of the structure group of an involutive solution to study the unique product property in such groups. An algorithm to find subgroups of a Bieberbach group isomorphic to the Promislow subgroup is introduced and then used in the case of structure group of involutive solutions. To extend the results related to retractability to non-involutive solutions, following the ideas of Meng, Ballester-Bolinches and Romero, we develop the theory of right $p$-nilpotent skew braces. The theory of left $p$-nilpotent skew braces is also developed and used to give a short proof of a theorem of Smoktunowicz in the context of skew braces.