Hopf Galois structures, skew braces for groups of size $p^nq$: The cyclic Sylow subgroup case

TL;DR

This paper classifies and enumerates Hopf-Galois structures and skew braces for groups of order p^n q with cyclic Sylow p-subgroups, providing explicit counts and a complete classification when q<p.

Contribution

It offers a detailed enumeration of Hopf-Galois structures and skew braces for groups of order p^n q with cyclic Sylow p-subgroups, including a complete classification in certain cases.

Findings

Enumerates Hopf-Galois structures for groups of order p^n q with cyclic Sylow p-subgroups.

Computes the number of skew braces with specified additive and multiplicative groups.

Provides a complete classification of these structures when q<p.

Abstract

Let $n\geq 1$ be an integer, $p$, $q$ be distinct odd primes. Let ${G}$, $N$ be two groups of order $p^nq$ with their Sylow-$p$-subgroups being cyclic. We enumerate the Hopf-Galois structures on a Galois ${G}$-extension, with type $N$. This also computes the number of skew braces with additive group isomorphic to $G$ and multiplicative group isomorphic to $N$. Further when $q<p$, we give a complete classification of the Hopf-Galois structures on Galois-$G$-extensions.