No abstract

Let be a group of order mp where p is prime and p > m. We give a strategy to enumerate the regular subgroups of Perm() normalized by the left representation λ() of. These regular subgroups are in one-to-one correspondence with the Hopf Galois structures on Galois field extensions L/K with = Gal(L/K). We prove that every such regular subgroup is contained in the normalizer in Perm() of the p-Sylow subgroup of λ(). This normalizer has an affine representation that makes feasible the explicit determination of regular subgroups in many cases. We illustrate our approach with a number of examples, including the cases of groups whose order is the product of two distinct primes and groups of order p(p − 1), where p is a "safe prime". These cases were previously studied by N. Byott and L. Childs, respectively.

The holomorph of a group G is Norm B G , the normalizer of the left regular representation G in its group of permutations B = Perm G . The multiple holomorph of G is the normalizer of the holomorph in B. The multiple holomorph and its quotient by the holomorph encodes a great deal of information about the holomorph itself and about the group G and its conjugates within the holomorph. We explore the multiple holomorphs of the dihedral groups D n and quaternionic (dicyclic) groups Q n for n ≥ 3.

Let L/K be a finite separable extension of fields whose Galois closure E/K has group G. Greither and Pareigis have used Galois descent to show that a Hopf algebra giving a Hopf-Galois structure on L/K has the form E[N ] G for some group N such that |N | = [L : K]. We formulate criteria for two such Hopf algebras to be isomorphic as Hopf algebras, and provide a variety of examples. In the case that the Hopf algebras in question are commutative, we also determine criteria for them to be isomorphic as K-algebras. By applying our results, we complete a detailed analysis of the distinct Hopf algebras and Kalgebras that appear in the classification of Hopf-Galois structures on a cyclic extension of degree p n , for p an odd prime number.

The work of Greither and Pareigis details the enumeration of the Hopf-Galois structures (if any) on a given separable field extension. We consider the cases where L/K is already classically Galois with Γ = Gal(L/K), where |Γ | = 4p for p > 3 a prime. The goal is to determine those regular (transitive and fixed point free) subgroups N of Perm(Γ ) that are normalized by the left regular representation of Γ . A key fact that aids in this search is the observation that any such regular subgroup, necessarily of order 4p, has a unique subgroup of order p. This allows us to show that all such N are contained in a 'twisted' wreath product, a subgroup of high index in Perm(Γ ) which has a very computationally convenient description that allows us to perform the aforementioned enumeration.