Classification of braces of cardinality $p^4$

TL;DR

This paper classifies all braces of size p^4 for large primes by analyzing nilpotent pre-Lie rings and their relation to braces, completing the classification for this size.

Contribution

It provides a complete classification of braces of size p^4, including nilpotent and non-nilpotent cases, for sufficiently large primes.

Findings

Nilpotent pre-Lie rings of size p^4 classified

Correspondence between pre-Lie rings and braces established

Complete classification of braces of size p^4 achieved

Abstract

We classify nilpotent pre-Lie rings of cardinality $p^4$ and thereby braces of the same cardinality, for a sufficiently large prime $p$. It has been shown that nilpotent pre-Lie rings of cardinality $p^n$ correspond to strongly nilpotent braces of the same cardinality, for sufficiently large $p$. These braces are explicitly obtained from the corresponding pre-Lie rings by the construction of the group of flows. Not right nilpotent braces of cardinality $p^4$ have been classified, hence our results finish the classification of braces of cardinality $p^4$.