AS-configurations and skew-translation generalised quadrangles

TL;DR

This paper investigates skew-translation generalized quadrangles (STGQ), providing new structural insights, classifications for specific orders, and extending understanding beyond known elementary abelian cases.

Contribution

It offers new structural results for groups with AS-configurations and classifies STGQ of order (8,8) and for odd q, expanding current knowledge.

Findings

Classified STGQ of order (8,8).

Extended classification of STGQ for odd q.

Provided new structural information for groups with AS-configurations.

Abstract

The only known skew-translation generalised quadrangles (STGQ) having order $(q,q)$, with $q$ even, are translation generalised quadrangles. Equivalently, the only known groups $G$ of order $q^3$, $q$ even, admitting an Ahrens-Szekeres (AS-)configuration are elementary abelian. In this paper we prove results in the theory of STGQ giving (i) new structural information for a group $G$ admitting an AS-configuration, (ii) a classification of the STGQ of order $(8,8)$, and (iii) a classification of the STGQ of order $(q,q)$ for odd $q$ (using work of Ghinelli and Yoshiara).