Beauville structures in finite p-groups

TL;DR

This paper investigates the existence and classification of Beauville structures in finite p-groups, extending known characterizations to broader classes and providing explicit counts and examples, including infinite families.

Contribution

It extends Catanese's characterization of Beauville groups to broader finite p-groups and determines explicit counts and examples, including infinite families.

Findings

Characterization applies to all p-groups of order at most p^p.

Exact number of Beauville groups of order p^5 and p^6 for certain primes.

Existence of infinite families of Beauville 3-groups for all n ≥ 5.

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 $p\ge 5$, and of order $p^6$, for $p\ge 7$. On the other hand, we determine which quotients of the Nottingham group over $\mathbb{F}_p$ 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 $n\ge 5$.