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Retractability of solutions to the Yang-Baxter equation and \(p\)-nilpotency of skew braces. (English) Zbl 1436.16044

T. Gateva-Ivanova and M. Van den Bergh [J. Algebra 206, No. 1, 97–112 (1998; Zbl 0944.20049)] proved that the structure group of a finite involutive set-theoretic solution to the Yang-Baxter equation is a Bieberbach group. In the paper under review, a relationship to the unique product property is investigated. A major role is played by Promislov’s 1988 counterexample of a torsion-free group where the unique product property does not hold [S. D. Promislow, Bull. Lond. Math. Soc. 20, No. 4, 302–304 (1988; Zbl 0662.20022)]. This suggests the strategy to check whether Promislov’s group occurs as a subgroup. An algorithm to check for subgroups of a Bieberbach group isomorphic to the Promislov group is designed. For finite involutive non-multipermutation solutions of the Yang-Baxter equation of order up to eight, it is decided with few exceptions when the unique product property is satisfied. For non-involutive solutions, recent results of Meng, Ballester-Bolinches, and Esteban-Romero [H. Meng et al., Proc. Edinb. Math. Soc., II. Ser. 62, No. 2, 595–608 (2019; Zbl 1471.17030)] are extended to skew-braces of nilpotent type.

MSC:

16T25 Yang-Baxter equations
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