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 $A\ast c=0$ for some central element $c$ of a brace $A$.