Retractability of solutions to the Yang Baxter equation and p nilpotency of skew braces
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 [Formula: see text] nilpotent skew braces. The theory of left [Formula: see text] nilpotent skew braces is also developed and used to give a short proof of a theorem of Smoktunowicz in the context of skew braces.
Int. J. Algebra Comput.