Simple groups, product actions, and generalised quadrangles

TL;DR

This paper advances the classification of flag-transitive generalised quadrangles by analyzing the structure of their automorphism groups, especially those preserving Cartesian product decompositions, and establishes bounds on the number of factors involved.

Contribution

It introduces new constraints on the automorphism groups of generalised quadrangles, particularly limiting the number of factors in Cartesian product decompositions under generic conditions.

Findings

Number of factors in Cartesian product decompositions is at most four.

G cannot have holomorph compound O'Nan-Scott type.

Poses new group-theoretic questions about simple groups and primitive permutation groups.

Abstract

The classification of flag-transitive generalised quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalised quadrangles are also point-primitive (up to point-line duality), it is likewise natural to seek a classification of the point-primitive examples. Working towards this aim, we are led to investigate generalised quadrangles that admit a collineation group $G$ preserving a Cartesian product decomposition of the set of points. It is shown that, under a generic assumption on $G$, the number of factors of such a Cartesian product can be at most four. This result is then used to treat various types of primitive and quasiprimitive point actions. In particular, it is shown that $G$ cannot have holomorph compound O'Nan-Scott type. Our arguments also pose purely group-theoretic questions about conjugacy classes in non-Abelian finite simple groups, and about fixities of primitive permutation groups.