Abelian Hopf Galois structures on prime-power Galois field extensions

The main theorem of this paper is that if (N, +) is a finite abelian p-group of p-rank m, where p > m+1, then every regular abelian subgroup of the holomorph of N is isomorphic to N. The proof utilizes a connection, observed by Caranti, Dalla Volta, and Sala, between regular abelian subgroups of the holomorph of N and nilpotent ring structures on (N, +). Examples are given that limit possible generalizations of the theorem. The primary application of the theorem is to Hopf Galois extensions of fields. Let L|K be a Galois extension of fields with abelian Galois group G. If also L|K is H-Hopf Galois, where the K-Hopf algebra H has associated group N with N as above, then N is isomorphic to G.