Hopf Galois structures, skew braces for groups of size \(p^n q\): the cyclic Sylow subgroup case (Q6635598)

Given two finite groups \(G\) and \(N\) of the same order, let \(e(G,N)\) denote the number of Hopf-Galois structures of type \(N\) on any finite Galois extension \(L/K\) with \(\mathrm{Gal}(L/K)\simeq G\). The determination of \(e(G,N)\) for different pairs \((G,N)\) of groups is an active line of research. A tool that is often used in the literature is the formula\N\[\Ne(G,N) = \frac{|\Aut(G)|}{|\Aut(N)|} \cdot e'(G,N)\N\]\Ndue to [\textit{C. Greither} and \textit{B. Pareigis}, J. Algebra 106, 239--258 (1987; Zbl 0615.12026)] and [\textit{N. P. Byott}, Commun. Algebra 24, No. 10, 3217--3228 (1996; Zbl 0878.12001)], which reduces the calculation of \(e(G,N)\) to a completely group-theoretic problem. Here \(e'(G,N)\) denotes the number of regular subgroups isomorphic to \(G\) in the holomorph\N\[\N\mathrm{Hol}(N) = N\rtimes \mathrm{Aut}(N)\N\]\Nof the group \(N\). In the paper under review, the authors continue this direction of research -- they computed \(e(G,N)\) and \(e'(G,N)\) for various pairs \((G,N)\) of groups of order \(p^nq\) (\(p,q\) are primes) having cyclic Sylow subgroups. Let us remark that the cases \(n=1,2\) are already known by [\textit{N. P. Byott}, J. Pure Appl. Algebra 188, No. 1--3, 45--57 (2004; Zbl 1047.16022)] and [\textit{E. Campedel} et al., J. Algebra 556, 1165--1210 (2020; Zbl 1465.12006)], respectively.