Characteristic subgroup lattices and Hopf–Galois structures

Abstract

The Hopf–Galois structures on normal field extensions [Formula: see text] with [Formula: see text] are in one-to-one correspondence with the set of regular subgroups [Formula: see text] of [Formula: see text], the group of permutations of [Formula: see text] as a set, that are normalized by the left regular representation [Formula: see text]. Each such [Formula: see text] corresponds to a Hopf algebra [Formula: see text] that acts on [Formula: see text]. Such regular subgroups need not be isomorphic to [Formula: see text] but must have the same order. One can divide all such [Formula: see text] into collections [Formula: see text], where [Formula: see text] is the set of those regular [Formula: see text] normalized by [Formula: see text] and isomorphic to a given abstract group [Formula: see text], where [Formula: see text]. There exists an injective correspondence between the characteristic subgroups of a given [Formula: see text] and the set of subgroups of [Formula: see text] stemming from the Galois correspondence between sub-Hopf algebras of [Formula: see text] and intermediate fields [Formula: see text], where [Formula: see text]. We utilize this correspondence to show that for certain pairs [Formula: see text], the collection [Formula: see text] must be empty. This also shows that for these [Formula: see text], there do not exist skew braces with additive group isomorphic to [Formula: see text] and circle group isomorphic to [Formula: see text].