Hopf-Galois structures on extensions of degree p2q and skew braces of order p2q: The cyclic Sylow p-subgroup case

Let p,q be distinct primes, with p>2. We classify the Hopf-Galois structures on Galois extensions of degree p2q, 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,⋅) of order p2q, in the case when the Sylow p-subgroups of G are cyclic. This is equivalent to classifying the skew braces (G,⋅,∘). Furthermore, we prove that if G and Γ are groups of order p2q with non-isomorphic Sylow p-subgroups, then there are no regular subgroups of the holomorph of G which are isomorphic to Γ. Equivalently, a Galois extension with Galois group Γ has no Hopf-Galois structures of type G. Our method relies on the alternate brace operation ∘ on G, which we use mainly indirectly, that is, in terms of the functions γ:G→Aut(G) defined by g↦(x↦(x∘g)⋅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 γ(gγ(h)⋅h)=γ(g)γ(h), for g,h∈G. We develop methods to deal with these functions, with the aim of making their enumeration easier, and more conceptual.