A Note on Beauville p-Groups

TL;DR

This paper classifies Beauville p-groups of small order and analyzes their prevalence among 2-generated groups, revealing that most are Beauville for order p^5 but not for p^6, and identifies the smallest non-abelian Beauville p-group for each prime.

Contribution

It provides a complete classification of Beauville p-groups up to order p^4 and studies their distribution among groups of order p^5 and p^6, including minimal examples.

Findings

All Beauville p-groups of order ≤ p^4 are classified.

The proportion of 2-generated groups of order p^5 that are Beauville approaches 1 as p increases.

For order p^6, the proportion of Beauville groups does not tend to 1.

Abstract

We examine which $p$-groups of order $\le p^6$ are Beauville. We completely classify them for groups of order $\le p^4$. We also show that the proportion of 2-generated groups of order $p^5$ which are Beauville tends to 1 as $p$ tends to infinity; this is not true, however, for groups of order $p^6$. For each prime $p$ we determine the smallest non-abelian Beauville $p$-group.