Purely (non-) strongly real Beauville $p$-groups

TL;DR

This paper constructs examples of Beauville p-groups that are never strongly real for primes p≥5, and identifies infinitely many purely strongly real Beauville 2-groups, addressing open questions in the field.

Contribution

It provides the first known examples of purely non-strongly real nilpotent Beauville groups for p≥5 and classifies infinitely many purely strongly real Beauville 2-groups.

Findings

Existence of purely non-strongly real Beauville p-groups for p≥5

Infinite families of purely strongly real Beauville 2-groups

Answers to questions posed by Fairbairn

Abstract

For every prime $p\geq5$, we give examples of Beauville $p$-groups whose Beauville structures are never strongly real. This shows that there are purely non-strongly real nilpotent Beauville groups. On the other hand, we determine infinitely many Beauville $2$-groups which are purely strongly real. This answers two questions formulated by Fairbairn in [8].