Simple groups, product actions, and generalised quadrangles
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 preserving a Cartesian product decomposition of the set of points. It is shown that, under a generic assumption on , 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 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.
Cite
@article{arxiv.1702.07308,
title = {Simple groups, product actions, and generalised quadrangles},
author = {John Bamberg and Tomasz Popiel and Cheryl E. Praeger},
journal= {arXiv preprint arXiv:1702.07308},
year = {2017}
}