Hopf-Galois structures on separable field extensions of degree \(pq\) (Q6117884)

Let \(L/K\) be a finite separable field extension with normal closure \(E\). Put \(G=\mathrm{Gal}(E/K)\) and \(G'=\mathrm{Gal}(E/L)\). For an abstract group \(N\) of order \([L:K]\), it is known by \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)] that Hopf-Galois structures on \(L/K\) correspond (non-bijectively) to the transitive subgroups \(M\) of the holomorph of \(N\) for which there is an isomoprhism \(M\rightarrow G\) taking \(\mathrm{Stab}_M(1_N)\) to \(G'\). Thus, the enumeration of Hopf-Galois structures on \(L/K\) can be reduced to a completely group-theoretic problem, and it has been studied in various settings throughout the years. The enumeration of Hopf-Galois structures on Galois extensions of squarefree degree was completed in [\textit{A. A. Alabdali} and \textit{N. P. Byott}, J. Algebra 559, 58--86 (2020; Zbl 1465.12005)]. The next natural step is to consider Hopf-Galois structures on arbitrary separable extensions of squarefree degree. However, the non-Galois case seems to be much more difficult. In the Galois case, one has \(|G|=|N| = [L:K]\), and groups of squarefree order have already been classified by [\textit{M. R. Murty} and \textit{V. K. Murty}, Math. Ann. 267, 299--309 (1984; Zbl 0531.10048)], so the situtation is fairly controllable. But in the non-Galois case, it need not be true that \(|G|\) is squarefree. As a first step to tackle this problem, \textit{N. P. Byott} and \textit{I. Martin-Lyons} [J. Pure Appl. Algebra 226, No. 3, Article ID 106869, 20 p. (2002; Zbl 1475.12006] assumed that \([L:K]=pq\) is a product of two distinct odd primes with \(p=2q+1\), and they classified the Hopf-Galois structures in this case. The paper under review extends their work by allowing \(p\) and \(q\) to be arbitrary distinct odd primes.