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 Γ, 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, +), 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 Γ 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.
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