The point regular automorphism groups of the Payne derived quadrangle of $W(q)$

TL;DR

This paper classifies the point regular automorphism groups of a specific finite geometric structure, the Payne derived quadrangle of $W(q)$, revealing that such groups can have unbounded nilpotency class.

Contribution

It completely determines the point regular automorphism groups of the Payne derived quadrangle of $W(q)$ for odd q, a new classification in finite geometry.

Findings

Automorphism groups can have unbounded nilpotency class.

Complete determination of point regular automorphism groups for the structure.

Implications for symmetry groups in finite generalized quadrangles.

Abstract

In this paper, we completely determine the point regular automorphism groups of the Payne derived quadrangle of the symplectic quadrangle $W(q)$, $q$ odd. As a corollary, we show that the finite groups that act regularly on the points of a finite generalized quadrangle can have unbounded nilpotency class.