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

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Higher Education Press 2022