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

Abstract

$\DeclareMathOperator{\Aut}{Aut}$Let $p,q$ be distinct primes, with $p>2$. We classify the Hopf-Galois structures on Galois extensions of degree $p^{2}q$, such that the Sylow $p$-subgroups of the Galois group are cyclic. This we do, according to Greither and Pareigis, and Byott, 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 cyclic. This is equivalent to classifying the skew braces $(G,\cdot ,\circ )$. Furthermore, we prove that if $G$ and $\Gamma$ are groups of order $p^{2}q$ with non-isomorphic Sylow $p$-subgroups, then there are no regular subgroups of the holomorph of $G$ which are isomorphic to $\Gamma$. Equivalently, a Galois extension with Galois group $\Gamma$ has no Hopf-Galois structures of type $G$. Our method relies on the alternate brace operation $\circ$ on $G$, which we use mainly indirectly, that is, in terms of the functions $γ: G \to \Aut(G)$ defined by $g\mapsto (x\mapsto (x\circ g)\cdot g^{-1})$. 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.

43 pages