Nilpotent groups of class three and braces

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] ⊆ Soc(H) and H/[H, H] is a standard abelian brace, where Soc(H) denotes the socle of the brace H.