Nilpotent and abelian Hopf–Galois structures on field extensions

Let L / K be a finite Galois extension of fields with group Γ. When Γ is nilpotent, we show that the problem of enumerating all nilpotent Hopf–Galois structures on L / K can be reduced to the corresponding problem for the Sylow subgroups of Γ. We use this to enumerate all nilpotent (resp. abelian) Hopf–Galois structures on a cyclic extension of arbitrary finite degree. When Γ is abelian, we give conditions under which every abelian Hopf–Galois structure on L / K has type Γ. We also give a criterion on n such that every Hopf–Galois structure on a cyclic extension of degree n has cyclic type.