Right Nilpotency of Braces of Cardinality $p^4$

TL;DR

This paper classifies the right nilpotency of braces with size p^4, showing all such braces are right nilpotent when their multiplicative group is non-abelian, using a specific algebraic condition.

Contribution

It provides a complete characterization of right nilpotency for braces of size p^4, especially for non-abelian multiplicative groups, using a new sufficient condition.

Findings

All braces of size p^4 with non-abelian groups are right nilpotent.

The condition A* c=0 for some central c guarantees right nilpotency.

Braces with abelian multiplicative groups are both left and right nilpotent.

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 $A$.