Hopf–Galois structures on parallel extensions

Let L / K be a finite separable extension of fields of degree n , and let E / K be its Galois closure. Greither and Pareigis showed how to find all Hopf–Galois structures on L / K . We will call a subextension L ′ / K of E / K parallel to L / K if [ L ′ : K ] = n . In this paper, we investigate the relationship between the Hopf–Galois structures on an extension L / K and those on the related parallel extensions. We give an example of a transitive subgroup corresponding to an extension admitting a Hopf–Galois structure but that has a parallel extension admitting no Hopf–Galois structures. We show that once one has such a situation, it can be extended into an infinite family of transitive subgroups admitting this phenomenon. We also investigate this fully in the case of extensions of degree pq with p , q distinct odd primes, and show that there is no example of such an extension admitting the phenomenon.