On the Galois correspondence for Hopf Galois structures

We study the question of the surjectivity of the Galois correspondence from subHopf algebras to subfields given by the Fundamental Theorem of Galois Theory for abelian Hopf Galois structures on a Galois extension of fields with Galois group G', a finite abelian p-group. Applying the connection between regular subgroups of the holomorph of a finite abelian p-group (G, +) and associative, commutative nilpotent algebra structures A on (G, +) of Caranti, et. al., we show that if A gives rise to a H-Hopf Galois structure on L/K, then the K-subHopf algebras of H correspond to the ideals of A. Among the applications, we show that if G and G' are both elementary abelian p-groups, then the only Hopf Galois structure on L/K of type G for which the Galois correspondence is surjective is the classical Galois structure on L/K.