## Abstract

New constructions of braces on finite nilpotent groups are given and

hence this leads to new solutions of the Yang-Baxter equation. In

particular, it follows that if a group $G$ of odd order is

nilpotent of class three, then it is a homomorphic image of the

multiplicative group of a finite left brace (i.e. an involutive Yang

Baxter group) which also is a nilpotent group of class three. We

give necessary and sufficient conditions for an arbitrary group $H$

to be the multiplicative group of a left brace such that $[H,H]

\subseteq \Soc (H)$ and $H/[H,H]$ is a standard abelian brace,

where $\Soc (H)$ denotes the socle of the brace $H$.

hence this leads to new solutions of the Yang-Baxter equation. In

particular, it follows that if a group $G$ of odd order is

nilpotent of class three, then it is a homomorphic image of the

multiplicative group of a finite left brace (i.e. an involutive Yang

Baxter group) which also is a nilpotent group of class three. We

give necessary and sufficient conditions for an arbitrary group $H$

to be the multiplicative group of a left brace such that $[H,H]

\subseteq \Soc (H)$ and $H/[H,H]$ is a standard abelian brace,

where $\Soc (H)$ denotes the socle of the brace $H$.

Original language | English |
---|---|

Pages (from-to) | 55-79 |

Number of pages | 16 |

Journal | Publicacions Matemàtiques |

Volume | 60 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2016 |

## Keywords

- Yang-Baxter equation
- set-theoretic solution
- brace
- nilpotent group
- metabelian group