Braces whose additive group has a cyclic maximal subgroup

Abstract

The problem of constructing all the non-degenerate involutive set theoretic solutions of the Yang-Baxter equation recently has been reduced to the problem of describing all the right braces. In particular, the classification of all finite right braces is fundamental in describing all such solutions of the Yang-Baxter equation. Let H be a right brace of order p^n, (H,+) \cong{\mathbb{Z}}_p\times {\mathbb{Z}}_{p^{n-1}}, where n⩾4 and p is odd prime. In this paper we prove \mathrm{S}\mathrm{o}\mathrm{c}\left(H\right)\ne 1 and classify all right braces H such that |\mathrm{S}\mathrm{o}\mathrm{c}\left(H\right)|=p^{n-1}.

Key words： Brace; finite p-group; Yang-Baxter equation

Acknowlegements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11771258, 11801341) and the Science Foundation of Shanxi Normal University (Grant No. 2019XSY019).

References

1 Bachiller D. Classification of braces of order p³. J Pure Appl Algebra 2015; 219(8): 3568–3603

2 Bachiller D, Cedó F, Jespers E. Solutions of the Yang-Baxter equation associated with a left brace. J Algebra 2016; 463: 80–102

3 Cedó F, Jespers E, Okniński J. Braces and the Yang-Baxter equation. Comm Math Phys 2014; 327(1): 101–116

4 Dietzel C. Braces of order p²q. J Algebra Appl 2021; 20(8): 2150140 (24 pp)

5 Rump W. Braces, radical rings, and the quantum Yang-Baxter equation. J Algebra 2007; 307(1): 153–170

6 Rump W. Classification of cyclic braces. J Pure Appl Algebra 2007; 209(3): 671–685

7 Rump W. Classification of cyclic braces, II. Trans Amer Math Soc 2019; 372(1): 305–328

8 XuM YQuH P. Finite p-Groups. Beijing: Beijing University Press, 2010 (in Chinese)