Skip to main content
Log in

Triply Factorised Groups and the Structure of Skew Left Braces

  • Published:
Communications in Mathematics and Statistics Aims and scope Submit manuscript

Abstract

The algebraic structure of skew left brace has proved to be useful as a source of set-theoretic solutions of the Yang–Baxter equation. We study in this paper the connections between left and right \(\pi \)-nilpotency and the structure of finite skew left braces. We also study factorisations of skew left braces and their impact on the skew left brace structure. As a consequence of our study, we define a Fitting-like ideal of a left brace. Our approach depends strongly on a description of a skew left brace in terms of a triply factorised group obtained from the action of the multiplicative group of the skew left brace on its additive group.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Acri, E., Lutowski, R., Vendramin, L.: Rectractability of solutions to the Yang–Baxter equation and $p$-nilpotency of skew braces. Int. J. Algebra Comput. 30(01), 91–115 (2020)

    Article  Google Scholar 

  2. Amberg, B., Franciosi, S., de Giovanni, F.: Products of Groups. Oxford Mathematical Monographs. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York (1992)

    MATH  Google Scholar 

  3. Bardakov, V.G., Neshchadim, M.V., Yadav, M.K.: Computing skew left braces of small orders. Int. J. Algebra Comput. 30(4), 839–851 (2020)

    Article  MathSciNet  Google Scholar 

  4. Byott, N.P.: Solubility criteria for Hopf–Galois structures. N. Y. J. Math. 21, 883–903 (2015)

    MathSciNet  MATH  Google Scholar 

  5. Cedó, F., Smoktunowicz, A., Vendramin, L.: Skew left braces of nilpotent type. Proc. Lond. Math. Soc. 118(6), 1367–1392 (2019)

    Article  MathSciNet  Google Scholar 

  6. Doerk, K., Hawkes, T.: Finite Soluble Groups. De Gruyter Expositions in Mathematics, vol. 4. Walter de Gruyter & Co., Berlin (1992)

    Book  Google Scholar 

  7. Drinfeld, V.G.: On some unsolved problems in quantum group theory. In: Kulish, P.P. (ed.) Quantum Groups. Proceedings of Workshops Held in the Euler International Mathematical Institute, Leningrad, fall 1990, volume 1510 of Lecture Notes in Mathematics, pp. 1–8. Springer, Berlin (1992)

  8. Guarnieri, L., Vendramin, L.: Skew-braces and the Yang–Baxter equation. Math. Comput. 86(307), 2519–2534 (2017)

    Article  MathSciNet  Google Scholar 

  9. Huppert, B.: Endliche Gruppen I. Grund, vol. 134. Math. Wiss. Springer, Berlin (1967)

  10. Jespers, E., Kubat, Ł, Van Antwerpen, A., Vendramin, L.: Factorizations of skew braces. Math. Ann. 375(3–4), 1649–1663 (2019)

    Article  MathSciNet  Google Scholar 

  11. Kassel, C.: Quantum Groups. Graduate Texts in Mathematics, vol. 155. Springer, New York (1995)

    Google Scholar 

  12. Meng, H., Ballester-Bolinches, A., Esteban-Romero, R.: Left braces and the quantum Yang–Baxter equation. Proc. Edinb. Math. Soc. 62(2), 595–608 (2019)

    Article  MathSciNet  Google Scholar 

  13. Nasybullov, T.: Connections between properties of the additive and the multiplicative groups of a two-sided skew brace. J. Algebra 540, 156–167 (2019)

    Article  MathSciNet  Google Scholar 

  14. Numata, M.: On the ${\pi }$-nilpotent length of ${\pi }$-solvable groups. Osaka J. Math. 8(3), 447–451 (1971)

    MathSciNet  MATH  Google Scholar 

  15. Rump, W.: Braces, radical rings, and the quantum Yang–Baxter equation. J. Algebra 307, 153–170 (2007)

    Article  MathSciNet  Google Scholar 

  16. Smoktunowicz, A., Vendramin, L.: On skew braces (with an appendix by N. Byott and L. Vendramin). J. Comb. Algebra 2(1), 47–86 (2018)

    Article  MathSciNet  Google Scholar 

  17. Sysak, Y.P.: Some examples of factorized gorups and their relation to ring theory. In: de Giovanni, F., et al. (eds.) Infinite Groups 1994. Proceedings of the International Conference, Ravello, Italy, May 23–27, 1994, pp. 257–269. Walter de Gruyter, Berlin (1996)

  18. Sysak, Y.P.: Products of groups and quantum Yang–Baxter equation. Notes of a talk in Advances in Group Theory and Applications, Porto Cesareo, Lecce, Italy (2011)

  19. The GAP Group: GAP—Groups, Algorithms, and Programming, Version 4.10.1. http://www.gap-system.org (2019)

  20. Vendramin, L., Konovalov, A.: YangBaxter: combinatorial solutions for the Yang–Baxter equation. Version 0.9.0. https://gap-packages.github.io/YangBaxter/ (2019)

  21. Yang, C.N.: Some exact results for many-body problem in one dimension with repulsive delta-function interaction. Phys. Rev. Lett 19, 1312–1315 (1967)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work has been supported by the research Grant PGC2018-095140-B-I00 from the Ministerio de Ciencia, Innovación y Universidades (Spanish Government), the Agencia Estatal de Investigación (Spain), and FEDER (European Union); and PROMETEO/2017/057 from the Generalitat (Valencian Community, Spain).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. Ballester-Bolinches.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ballester-Bolinches, A., Esteban-Romero, R. Triply Factorised Groups and the Structure of Skew Left Braces. Commun. Math. Stat. 10, 353–370 (2022). https://doi.org/10.1007/s40304-021-00239-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40304-021-00239-6

Keywords

Mathematics Subject Classification

Navigation