## 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

Zbl 1371.16037

### Software:

Magma; GAP; SmallGrp
Full Text:

### References:

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