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}.

Keywords

Brace / finite p-group / Yang-Baxter equation

References

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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).

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