Braces of order $p^2q$

TL;DR

This paper classifies all left braces of order p^2q for primes p,q with q > p+1, proving three conjectures about their isomorphism classes and advancing understanding of algebraic structures related to these orders.

Contribution

It provides a complete classification of left braces of order p^2q and proves three conjectures regarding their isomorphism classes for certain prime values.

Findings

Classification of left braces of order p^2q

Proof of three conjectures by Guarnieri and Vendramin

Determination of the number of isomorphism classes for specific primes

Abstract

In this article we classify the left braces of order $p^2q$ where $p,q$ are primes fulfilling $q > p+1$. This classification includes a proof of three conjectures of Guarnieri and Vendramin (\cite[Conjectures 6.2-6.4]{Vendramin_skew}) concerning the number of isomorphism classes of left braces of order $p^2q$ for certain values of $p,q$.