Skew Braces of Squarefree Order

Ali A. Alabdali; Nigel P. Byott

Skew Braces of Squarefree Order

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arxiv 1910.07814 v2 pith:SGSGJ776 submitted 2019-10-17 math.RA math.QA

Ali A. Alabdali , Nigel P. Byott This is my paper · ORCID

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keywords bracesorderskewsquarefreegroupadditiveapplicationdegree

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abstract

Let $n \geq 1$ be a squarefree integer, and let $M$, $A$ be two groups of order $n$. Using our previous results on the enumeration of Hopf-Galois structures on Galois extensions of fields of squarefree degree, we determine the number of skew braces (up to isomorphism) with multiplicative group $M$ and additive group $A$. As an application, we enumerate skew braces whose order is the product of three distinct primes.

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Skew braces of size $pq$
math.GR 2019-08 accept novelty 6.0 of 10

For primes p>q, all skew braces of order pq are explicitly constructed: only the trivial brace if p is not 1 modulo q, and exactly 2q+2 braces if p is 1 modulo q.

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