A note on set-theoretic solutions of the Yang-Baxter equation. (English) Zbl 1442.16037
Summary: This paper shows that every finite non-degenerate involutive set theoretic solution \((X, r)\) of the Yang-Baxter equation whose permutation group \(\mathcal{G}(X, r)\) has cardinality which is a cube-free number is a multipermutation solution. Some properties of finite braces are also investigated. It is also shown that if \(A\) is a left brace whose cardinality is an odd number and \((- a) \cdot b = -(a \cdot b)\) for all \(a, b \in A\), then \(A\) is a two-sided brace and hence a Jacobson radical ring. It is also observed that the semidirect product and the wreath product of braces of a finite multipermutation level is a brace of a finite multipermutation level.
MSC:
| 16T25 | Yang-Baxter equations |
| 20D15 | Finite nilpotent groups, \(p\)-groups |
| 16N80 | General radicals and associative rings |
| 16N20 | Jacobson radical, quasimultiplication |
Keywords:
Jacobson radical ring; braces; braided groups; the Yang-Baxter equation; multipermutation solutionsReferences:
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