Hopf–Galois structures arising from groups with unique subgroup of order $p$

Abstract

For $\Gamma$ a group of order $mp$, where $p$ is a prime with $gcd\left(p,m\right)=1$, we consider the regular subgroups $N\le Perm\left(\Gamma \right)$ that are normalized by $\lambda \left(\Gamma \right)$, 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 $\Gamma =Gal\left(L∕K\right)$. Elsewhere we showed that if $p>m$ then all such $N$ lie within the normalizer of the Sylow $p$-subgroup of $\lambda \left(\Gamma \right)$. Here we show that one only need assume that all groups of a given order $mp$ have a unique Sylow $p$-subgroup, and that $p$ not be a divisor of the order of the automorphism groups of any group of order $m$. We thus extend the applicability of the program for computing these regular subgroups $N$ and concordantly the corresponding Hopf–Galois structures on separable extensions of degree $mp$.