Simple groups, product actions, and generalised quadrangles

Research output: Contribution to journalArticle

1 Citation (Scopus)
10 Downloads (Pure)

Abstract

The classification of flag-transitive generalized quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalized quadrangles are also point-primitive (up to point–line duality), it is likewise natural to seek a classification of the point-primitive examples. Working toward this aim, we are led to investigate generalized 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 nonabelian finite simple groups and fixities of primitive permutation groups.
Original languageEnglish
Pages (from-to)87-126
JournalNagoya Mathematical Journal
Volume234
Early online date14 Sep 2017
DOIs
Publication statusPublished - Jun 2019

Fingerprint

Generalized Quadrangle
Simple group
Flag-transitive
Permutation group
Cartesian product
Primitive Group
Finite Geometry
Collineation
Finite Simple Group
Group Theory
Conjugacy class
Set of points
Open Problems
Duality
Decompose

Cite this

@article{7204a0a535b343e58e3ab2c626fed6a5,
title = "Simple groups, product actions, and generalised quadrangles",
abstract = "The classification of flag-transitive generalized quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalized quadrangles are also point-primitive (up to point–line duality), it is likewise natural to seek a classification of the point-primitive examples. Working toward this aim, we are led to investigate generalized 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 nonabelian finite simple groups and fixities of primitive permutation groups.",
keywords = "Mathematics subject classi cation. Primary 51E12; Secondary 20B15, 05B25",
author = "John Bamberg and Tomasz Popiel and Praeger, {Cheryl Elisabeth}",
year = "2019",
month = "6",
doi = "10.1017/nmj.2017.35",
language = "English",
volume = "234",
pages = "87--126",
journal = "Nagoya Mathematical Journal",
issn = "0027-7630",
publisher = "Cambridge University Press",

}

Simple groups, product actions, and generalised quadrangles. / Bamberg, John; Popiel, Tomasz; Praeger, Cheryl Elisabeth.

In: Nagoya Mathematical Journal, Vol. 234, 06.2019, p. 87-126.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Simple groups, product actions, and generalised quadrangles

AU - Bamberg, John

AU - Popiel, Tomasz

AU - Praeger, Cheryl Elisabeth

PY - 2019/6

Y1 - 2019/6

N2 - The classification of flag-transitive generalized quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalized quadrangles are also point-primitive (up to point–line duality), it is likewise natural to seek a classification of the point-primitive examples. Working toward this aim, we are led to investigate generalized 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 nonabelian finite simple groups and fixities of primitive permutation groups.

AB - The classification of flag-transitive generalized quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalized quadrangles are also point-primitive (up to point–line duality), it is likewise natural to seek a classification of the point-primitive examples. Working toward this aim, we are led to investigate generalized 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 nonabelian finite simple groups and fixities of primitive permutation groups.

KW - Mathematics subject classication. Primary 51E12; Secondary 20B15, 05B25

U2 - 10.1017/nmj.2017.35

DO - 10.1017/nmj.2017.35

M3 - Article

VL - 234

SP - 87

EP - 126

JO - Nagoya Mathematical Journal

JF - Nagoya Mathematical Journal

SN - 0027-7630

ER -