On the multipliers of a Singer quadrangle

TL;DR

This paper introduces the concept of multipliers in Singer quadrangles, explores their properties, and applies this to prove that certain automorphism groups cannot have a specific O'Nan-Scott type, resolving an open problem.

Contribution

It defines multipliers for Singer quadrangles and uses this to analyze automorphism groups, answering an open question about their possible types.

Findings

Multipliers are introduced and their properties are studied.

A point-primitive automorphism group cannot have O'Nan-Scott type HS.

The result addresses an open problem in the theory of generalized quadrangles.

Abstract

A finite generalized quadrangle $\cS$ is a Singer quadrangle if it has an automorphism group that acts sharply transitively on its points. In this paper, we introduce the notion of multipliers for a Singer quadrangle and study their basic properties. As an application, we show that a point-primitive automorphism group of a thick generalized quadrangle cannot have O'Nan-Scott type HS (holomorph simple), which answers an open problem in \cite{Bamberg 2019}.