Elation generalised quadrangles of order (s,p), where p is prime

TL;DR

This paper characterizes elation generalized quadrangles of order (s,p) with p prime, showing they are either classical or derived from a flock of a quadratic cone, thus classifying their structure.

Contribution

It provides a classification of elation generalized quadrangles of order (s,p) with p prime, identifying their origins as classical or flock quadrangles.

Findings

Elation generalized quadrangles with p+1 lines per point are either classical or flock quadrangles.

The structure is fully characterized for prime p.

The result advances understanding of the geometric and algebraic properties of these quadrangles.

Abstract

We show that an elation generalised quadrangle which has p+1 lines on each point, for some prime p, is classical or arises from a flock of a quadratic cone (i.e., is a flock quadrangle).