An infinite family of strongly real Beauville $p$-groups

TL;DR

This paper constructs an infinite family of non-abelian strongly real Beauville p-groups for all primes p, using quotients of triangle groups, and determines their existence for various orders.

Contribution

It introduces a new method to generate strongly real Beauville p-groups for all primes p and all sufficiently large orders.

Findings

Existence of non-abelian strongly real Beauville p-groups for all primes p.

Construction of infinite families of such groups via triangle group quotients.

Strongly real Beauville p-groups exist precisely at the same orders as Beauville p-groups.

Abstract

We give an infinite family of non-abelian strongly real Beauville $p$-groups for every prime $p$ by considering the quotients of triangle groups, and indeed we prove that there are non-abelian strongly real Beauville $p$-groups of order $p^n$ for every $n \geq 3, 5$ or $7$ according as $p \geq 5$ or $p =3$ or $p =2$. This shows that there are strongly real Beauville $p$-groups exactly for the same orders for which there exist Beauville $p$-groups.