Point regular groups of automorphisms of generalised quadrangles

TL;DR

This paper classifies and constructs point regular automorphism groups of known generalized quadrangles, including classical and derived types, revealing diverse group structures and enumerating possibilities for small cases.

Contribution

It determines all point regular automorphism groups of classical generalized quadrangles and constructs new examples for derived quadrangles, highlighting diverse group structures.

Findings

Classified all point regular automorphism groups of classical generalized quadrangles.

Constructed new point regular groups for derived quadrangles of order (q-1,q+1).

Enumerated all such groups for some small generalized quadrangles.

Abstract

We study the point regular groups of automorphisms of some of the known generalised quadrangles. In particular we determine all point regular groups of automorphisms of the thick classical generalised quadrangles. We also construct point regular groups of automorphisms of the generalised quadrangle of order $(q-1,q+1)$ obtained by Payne derivation from the classical symplectic quadrangle $\mathsf{W}(3,q)$. For $q=p^f$ with $f\geq 2$ we obtain at least two nonisomorphic groups when $p\geq 5$ and at least three nonisomorphic groups when $p=2$ or $3$. Our groups include nonabelian 2-groups, groups of exponent 9 and nonspecial $p$-groups. We also enumerate all point regular groups of automorphisms of some small generalised quadrangles.