Publicacions Matemàtiques

Nilpotent Groups of Class Three and Braces

Ferran Cedó, Eric Jespers, and Jan OkniŃski

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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 $[H,H] \subseteq \operatorname{Soc} (H)$ and $H/[H,H]$ is a standard abelian brace, where $\operatorname{Soc} (H)$ denotes the socle of the brace $H$.

Article information

Publ. Mat. Volume 60, Number 1 (2016), 55-79.

First available in Project Euclid: 22 December 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 16T25: Yang-Baxter equations 20F18: Nilpotent groups [See also 20D15] 20F16: Solvable groups, supersolvable groups [See also 20D10]

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


Cedó, Ferran; Jespers, Eric; OkniŃski, Jan. Nilpotent Groups of Class Three and Braces. Publ. Mat. 60 (2016), no. 1, 55--79.

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