Beauville structures in finite p-groups

Abstract

We study the existence of (unmixed) Beauville structures in finite p-groups, where p is a prime. First of all, we extend Catanese's characterisation of abelian Beauville groups to finite p-groups satisfying certain conditions which are much weaker than commutativity. This result applies to all known families of p-groups with a good behaviour with respect to powers: regular p-groups, powerful p-groups and more generally potent p-groups, and (generalised) p-central p-groups. In particular, our characterisation holds for all p-groups of order at most pp, which allows us to determine the exact number of Beauville groups of order p5, for p≥5, and of order p6, for p≥7. On the other hand, we determine which quotients of the Nottingham group over Fp are Beauville groups, for an odd prime p. As a consequence, we give the first explicit infinite family of Beauville 3-groups, and we show that there are Beauville 3-groups of order 3n for every n≥5.