Hopf-Galois structures on extensions of degree p2q and skew braces of order p2q: the elementary abelian Sylow p-subgroup case

Let p, q be distinct primes, with p > 2. In a previous paper we classified the Hopf-Galois structures on Galois extensions of degree p2q, when the Sylow p-subgroups of the Galois group are cyclic. This is equivalent to classifying the skew braces of order p2q, for which the Sylow p-subgroups of the multiplicative group are 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, for the groups (G, ·) of order p2q, the regular subgroups of their holomorphs whose Sylow p-subgroups are elementary abelian. We rely on the use of certain gamma functions γ: G → 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 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.