Classification of Uniconnected Involutive Solutions of the Yang–Baxter Equation With Odd Size and a Z-Group Permutation Group

Abstract

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.