  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.

Rings and Algebras;Group Theory;Quantum Algebra