Classification of braces of order $p^3$

TL;DR

This paper classifies all left braces of order p^3 for any prime p by building on classifications of smaller orders and explicitly constructing the necessary algebraic structures.

Contribution

It provides a complete classification of left braces of order p^3, extending previous classifications of smaller orders and explicitly constructing the required algebraic operations.

Findings

Complete classification of left braces of order p^3

Explicit construction methods for left braces

Extension of classifications from order p and p^2

Abstract

A classification up to isomorphism of all left braces of order $p^3$, where $p$ is any prime number, is given. To this end, we first classify all the left braces of order $p$ and $p^2$, and then we construct explicitly the hypothesis required in Corollary D of N. Ben David's Ph.D. thesis to build multiplications of left braces.