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Computing skew left braces of small orders. (English) Zbl 1458.16040

A skew brace it the datum of two group operations \(+\) and \(\circ\) on the same set \(G\), satisfying the twisted distributivity relation \[a\circ (b+c) = a \circ b - a + a \circ c.\] They play a central role in the classification of set-theoretic solutions to the Yang-Baxter equation, and are closely connected to Hopf-Galois structures.
Besides introducing this notion, the pioneer paper [L. Guarnieri and L. Vendramin, Math. Comput. 86, No. 307, 2519–2534 (2017; Zbl 1371.16037)] suggested an algorithm for classifying skew braces of small size, which was used in practice up to size \(168\) with some exceptions. The main step of this algorithm is the computation, for a finite group \((G,+)\), of all regular subgroups \(R\) of its holomorph \(\operatorname{Hol}(G)\) of order \(|G|\) up to conjugation by elements of \(\operatorname{Aut}(G)\). The second group operation \(\circ\) on \(G\) is then extracted from \(R\).
The main observation of the present paper is that in the above algorithm, it suffices to consider subgroups up to conjugation by the whole \(\operatorname{Hol}(G)\). This considerably reduces the computational complexity, and allows the authors to go up to size \(868\) except for certain cases (mainly the multiples of \(32\)). The authors also improve the algorithm for \(p\)-groups. Finally, observing the data obtained, they propose conjectures for particular types of skew brace size: \(p^2q\) and \(2pq\) for prime \(q>p\), \(8p\) and \(12p\) for prime \(p\). Some of these conjectures were recently confirmed.

MSC:

16T25 Yang-Baxter equations

Citations:

Zbl 1371.16037

Software:

Magma; GAP; SmallGrp
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Full Text: DOI arXiv

References:

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