Abelian Hopf Galois structures on prime-power Galois field extensions

Abstract:

The main theorem of this paper is that if $(N, +)$ is a finite abelian $p$-group of $p$-rank $m$ where $m+1< p$, 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 $G$.