On the Galois correspondence for Hopf-Galois structures

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