Braces were introduced by [W. Rump, J. Algebra 307, 153–170 (2007; Zbl 1115.16022)] as a tool to study non-degenerate set-theoretic involutive solutions of the Yang-Baxter equation, and were generalized to skew braces by [L. Guarnieri and L. Vendramin, Math. Comput. 86, No. 307, 2519–2534 (2017; Zbl 1371.16037)] in order to capture the non-involutive solutions as well. Here, a skew brace is a triplet \((B,+,\circ)\) for which \((B,+)\) and \((B,\circ)\) are groups satisfying \[ a \circ (b + c) = (a\circ b) - a + (a\circ c)\mbox{ for all }a,b,c\in B. \] Despite the notation, the group \((B,+)\) is not assumed to be abelian in general. A brace is precisely a skew brace \((B,+,\circ)\) for which \((B,+)\) is abelian. Due to their connection with the Yang-Baxter equation, the classification of the isomorphism classes of skew braces is a problem of interest.
In the paper under review, the authors enumerated skew braces \((B,+,\circ)\) of order \(p^2q\) with non-abelian \((B,+)\) for distinct primes \(p,q\). It was done using the construction of skew braces developed by Guarnieri and Vendramin in the aforementioned paper, where it was shown that isomorphism classes of skew braces \((B,+,\circ)\) are parametrized by the orbits of regular subgroups of \(\mathrm{Hol}(B,+)\) under conjugation by \(\operatorname{Aut}(B,+)\). Here \[ \mathrm{Hol}(B,+) = (B,+)\rtimes \operatorname{Aut}(B,+) \] denotes the holomorph of \((B,+)\). The authors found all such orbits by sorting the regular subgroups based on the size of their projection onto \(\operatorname{Aut}(B,+)\).
In Part I [Algebra Colloq. 29, 297–320 (2022; Zbl 1495.16030)], using the same approach, the same authors enumerated (skew) braces \((B,+,\circ)\) of order \(p^2q\) with abelian \((B,+)\). Thus, the classification of skew braces of order \(p^2q\) for distinct primes \(p,q\) is complete.
Remark. The enumeration of skew braces of order \(p^2q\) was achieved independently by E. Campedel in her PhD thesis [Hopf-Galois structures and skew braces of order \(p^2 q\). Milan: University of Milano-Bicocca (PhD Thesis) (2022)]. It was also done by considering regular subgroups of the holomorph, but she investigated them using these so-called gamma functions introduced by [A. Caranti and F. Dalla Volta, J. Algebra 507, 81–102 (2018; Zbl 1418.20008)]. Her thesis also provides counts of the corresponding Hopf-Galois structures, which are known to be related to regular subgroups of the holomorph. The results for groups of order \(p^2q\) having cyclic Sylow \(p\)-subgroups are published in [E. Campedel, A. Caranti, and I. Del Corso, J. Algebra 556, 1165–120 (2020; Zbl 1465.12006)].