On the asymptotic behaviour of the number of Beauville and non-Beauville $p$-groups

TL;DR

This paper establishes that both Beauville and non-Beauville 2-generator finite p-groups are asymptotically abundant within the family of finite p-groups, matching the known lower bounds for all such groups.

Contribution

It provides asymptotic lower bounds for the counts of Beauville and non-Beauville p-groups, showing their abundance among finite p-groups.

Findings

Both Beauville and non-Beauville groups are asymptotically abundant.

The bounds match the best known bounds for all 2-generator finite p-groups.

Beauville groups constitute a significant portion of 2-generator p-groups.

Abstract

We find asymptotic lower bounds for the numbers of both Beauville and non-Beauville $2$-generator finite $p$-groups of a fixed order, which turn out to coincide with the best known asymptotic lower bound for the total number of $2$-generator finite $p$-groups of the same order. This shows that both Beauville and non-Beauville groups are abundant within the family of finite $p$-groups.