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 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, .) 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, ., circle). 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 circle on G, which we use mainly indirectly, that is, in terms of the functions gamma : G -> Aut(G) defined by g bar right arrow (x bar right arrow (x circle 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 gamma(g(gamma(h)) . h) = gamma(g)gamma(h), for g, h is an element of G. We develop methods to deal with these functions, with the aim of making their enumeration easier, and more conceptual. (C) 2020 Elsevier Inc. All rights reserved.