Classification of uniconnected involutive solutions of the Yang-Baxter equation with odd size and a Z-group permutation group

TL;DR

This paper classifies certain finite involutive solutions to the Yang-Baxter equation using left braces, focusing on solutions with odd size and Z-group permutation groups, providing a comprehensive structural understanding.

Contribution

It introduces a classification of uniconnected involutive solutions with odd size and Z-group permutation groups using the framework of left braces.

Findings

Classification of solutions with odd size and Z-group permutation groups

Description of retraction of solutions via left braces

Complete classification of solutions with odd square-free size

Abstract

In the first part of this paper, we investigate the retraction of finite uniconnected involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation by means of left braces, giving a precise description in some cases. In the core of the paper, we also use left braces to classify all the uniconnected involutive non-degenerate set-theoretic solutions having odd size and a Z-group permutation group. As an application, we classify all the uniconnected involutive non-degenerate solutions having odd square-free size.