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 p p , which allows us to determine the exact number of Beauville groups of order p 5 , for p ≥ 5, and of order p 6 , 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 3 n for every n ≥ 5. [6,10,21]. We briefly mention some results that we want to highlight. In

On the asymptotic behaviour of the number of Beauville and non‐Beauville p‐groups

“…Then, for all m, n ∈ N, both [P n (G), P m (G)] and P n (G) p m are contained in P n+m (G). Now we can translate (6) and (7) into congruences with respect to subgroups of the lower p-central series.…”

“…(ii) Now we apply (7) to [a p m , g] with a ∈ P n (G) and g ∈ G. We only have to observe that, if K = a, [a, g] , then…”

“…In particular, it holds for ‐groups of nilpotency class less than . On the other hand, examples are given in showing that the above criterion given by is not generally valid without assumption , and it may be very hard to decide whether a given finite ‐group has a Beauville structure.…”

“…The determination of nilpotent Beauville groups is easily reduced to the case of p-groups. In [7], the first two authors found a criterion for a finite p-group with a 'nice power structure' to be a Beauville group that extends Catanese's condition for abelian groups. More precisely, if G is a 2-generator finite p-group of exponent p e that satisfies the condition…”

“…Note that condition (2) holds for most of the usual families of p-groups such as powerful groups, p-central groups, and regular p-groups. In particular, it holds for p-groups of nilpotency class less than p. On the other hand, examples are given in [7] showing that the above criterion given by (3) is not generally valid without assumption (2), and it may be very hard to decide whether a given finite p-group has a Beauville structure. Failing to find a general recipe to tell apart Beauville p-groups from non-Beauville ones, we can reformulate our goal and try to determine the asymptotic behaviour of the number of Beauville groups.…”