NILPOTENT GROUPS OF CLASS THREE AND BRACES

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