Skew braces of size $pq$

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 $p\not\equiv 1(\mathrm{mod}q)$ there exists only one skew brace which is the trivial one. When $p\equiv 1(\mathrm{mod}q)$, 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.

9 pages. Final version, accepted for publication in Communications in Algebra. arXiv admin note: text overlap with arXiv:1912.11889