# Skew braces of size $pq$

## TL;DR

This paper classifies all skew braces of size pq, where p and q are primes, using Hopf-Galois extension theory, revealing the structure and number of such braces depending on prime congruences.

## Contribution

It provides a complete classification of skew braces of size pq, connecting them with Hopf-Galois extensions and detailing their types based on prime congruences.

## Key findings

- Only the trivial skew brace exists when p ≡ 1 mod q.
- There are 2q+2 skew braces when p ≡ 1 mod q.
- Among these, 2 are cyclic, and 2q are non-abelian.

## Abstract

We construct all skew braces of size $pq$ (where $p>q$ are primes) by using Byott's classification of Hopf--Galois extensions of the same degree. For $p\not\equiv 1 \pmod{q}$ there exists only one skew brace which is the trivial one. When $p\equiv 1 \pmod{q}$, we have $2q+2$ skew braces, two of which are of cyclic type (so, contained in Rump's classification) and $2q$ of non-abelian type.