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.

arXiv comment9 pages. Final version, accepted for publication in Communications in Algebra. arXiv admin note: text overlap with arXiv:1912.11889

References in corpus (4)

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• On set-theoretical solutions of the quantum Yang-Baxter equation
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• On skew braces (with an appendix by N. Byott and L. Vendramin)

Cited by in corpus (8)

• Set-theoretic solutions of the Yang-Baxter equation associated to weak braces
• Inverse semi-braces and the Yang-Baxter equation
• Nilpotency in left semi-braces
• Extensions and Well's type exact sequence of skew braces
• Skew braces of size $p^{2}q$ II: non-abelian type
• Affine structures on groups and semi-braces
• Common divisor graphs for skew braces
• Relative Rota-Baxter operators of weight 0 on groups, pre-groups, braces, the Yang-Baxter equation and $T$-structures