Hopf-Galois Structures Arising From Groups with Unique Subgroup of Order p

TL;DR

This paper extends the classification of Hopf-Galois structures on degree $mp$ extensions by relaxing previous assumptions, focusing on groups with a unique p-Sylow subgroup and automorphism group conditions.

Contribution

It generalizes earlier results by showing that the classification holds under broader conditions, enabling more comprehensive computation of Hopf-Galois structures.

Findings

Extended the classification to groups with a unique p-Sylow subgroup

Reduced restrictions on automorphism groups of order m

Enhanced methods for computing Hopf-Galois structures

Abstract

For $\Gamma$ a group of order $mp$ for $p$ prime where $gcd(p,m)=1$, we consider those regular subgroups $N\leq Perm(\Gamma)$ normalized by $\lambda(\Gamma)$, the left regular representation of $\Gamma$. These subgroups are in one-to-one correspondence with the Hopf-Galois structures on separable field extensions $L/K$ with $\Gamma=Gal(L/K)$. This is a follow up to the author's earlier work where, by assuming $p>m$, one has that all such $N$ lie within the normalizer of the $p$-Sylow subgroup of $\lambda(\Gamma)$. Here we show that one only need assume that all groups of a given order $mp$ have a unique $p$-Sylow subgroup, and that $p$ not be a divisor of the automorphism groups of any group of order $m$. As such, we extend the applicability of the program for computing these regular subgroups $N$ and concordantly the corresponding Hopf-Galois structures on separable extensions of degree $mp$.