Hopf-Galois structures on extensions of degree $p^{2} q$ and skew braces of order $p^{2} q$: the elementary abelian Sylow $p$-subgroup case

TL;DR

This paper completes the classification of Hopf-Galois structures and skew braces of order p^2 q by analyzing elementary abelian Sylow p-subgroups, using gamma functions to classify regular subgroups of holomorphs.

Contribution

It extends previous work by classifying cases with elementary abelian Sylow p-subgroups using gamma functions and holomorph subgroup analysis.

Findings

Complete classification of Hopf-Galois structures for elementary abelian Sylow p-subgroups.

Development of methods to analyze gamma functions for subgroup enumeration.

Enhanced understanding of skew braces of order p^2 q.

Abstract

Let $p, q$ be distinct primes, with $p > 2$. In a previous paper we classified the Hopf-Galois structures on Galois extensions of degree $p^{2} q$, when the Sylow $p$-subgroups of the Galois group are cyclic. This is equivalent to classifying the skew braces of order $p^2q$, for which the Sylow $p$-subgroups of the multiplicative group is cyclic. In this paper we complete the classification by dealing with the case when the Sylow $p$-subgroups of the Galois group are elementary abelian. According to Greither and Pareigis, and Byott, we will do this by classifying the regular subgroups of the holomorphs of the groups $(G, \cdot)$ of order $p^{2} q$, in the case when the Sylow $p$-subgroups of $G$ are elementary abelian. We rely on the use of certain gamma functions $\gamma:G\to \operatorname{Aut}(G)$. These functions are in one-to-one correspondence with the regular subgroups of the holomorph of $G$, and are characterised by the functional equation $\gamma(g^{\gamma(h)} \cdot h) = \gamma(g) \gamma(h)$, for $g, h \in G$. We develop methods to deal with these functions, with the aim of making their enumeration easier and more conceptual.