Automatic realization of Hopf Galois structures

TL;DR

This paper investigates the structure and uniqueness of Hopf Galois structures on certain separable field extensions of degree p^n, establishing conditions for their abelian types and relationships with nonabelian groups.

Contribution

It proves the uniqueness of abelian Hopf Galois structures when p > n and relates nonabelian structures to abelian ones with matching element orders.

Findings

At most one abelian Hopf Galois structure when p > n.

Existence of a corresponding abelian structure for nonabelian types with specific properties.

Constraints on the types of Hopf Galois structures for degree p^n extensions.

Abstract

We consider Hopf Galois structures on a separable field extension $L/K$ of degree $p^n$, for $p$ an odd prime number, $n\geq 3$. For $p > n$, we prove that $L/K$ has at most one abelian type of Hopf Galois structures. For a nonabelian group $N$ of order $p^n$, with commutator subgroup of order $p$, we prove that if $L/K$ has a Hopf Galois structure of type $N$, then it has a Hopf Galois structure of type $A$, where $A$ is an abelian group of order $p^n$ and having the same number of elements of order $p^m$ as $N$, for $1\leq m \leq n$.