Right Nilpotency of Braces of Cardinality $p^4$

Abstract

We determine right nilpotency of braces of cardinality p 4 . If a brace of cardinality p 4 has an abelian multiplicative group, then it is left and right nilpotent, so we only consider braces with non-abelian multiplicative groups. We show right nilpotency in all cases using the sufficient condition of A * c = 0 for some central element c of a brace A.

On the passage from finite braces to pre-Lie rings

“…Right nilpotent braces have associated set-theoretic solutions of the Yang-Baxter equation of finite multipermutation level, and this class of set-theoretic solutions was investigated by several authors [13,23]. The right nilpotency of braces and skew braces has also been investigated [40,15,38]. In the last section, we translate some results on right nilpotent braces to right nilpotent pre-Lie algebras.…”