On a geometry of Ivanov and Shpectorov for the O'Nan sporadic simple group
type: Item
instance of: scholarly article
title: On a geometry of Ivanov and Shpectorov for the O'Nan sporadic simple group
zbMATH Open document ID: 0930.20014
author: Dimitri Leemans
author: Francis Buekenhout
published in: Journal of Combinatorial Theory. Series A
publication date: 1999-11-24T00:00:00Z
review text: The paper contributes to a project aiming to classify all geometries admitting flag-transitive actions of finite simple groups satisfying the properties \((\text{IP})_2\) (every rank 2 residue is either a partial linear space or a generalized digon) and RWPRI (for every flag its stabilizer acts primitively on the set of elements of at least one type in the corresponding residue). It is shown in the paper that for the O'Nan sporadic simple group \(O'N\) there are no \((\text{IP})_2\) and RWPRI geometries of rank 6 with \(M_{11}\) as a maximal parabolic or of rank 5 with \(J_1\) as a maximal parabolic. A geometry \({\mathcal G}(O'N)\) of rank 5 for \(O'N\) was constructed by \textit{A. A. Ivanov} and \textit{S. V. Shpektorov} [Usp. Mat. Nauk 41, No. 3(249), 183-184 (1986; Zbl 0626.20009)] (it has both \(M_{11}\) and \(J_1\) as maximal parabolics). By the result in the paper \({\mathcal G}(O'N)\) is not RWPRI. Up to the best of the reviewer's knowledge another important question about \({\mathcal G}(O'N)\) is unanswered: it is not known whether or not the triple cover of \({\mathcal G}(O'N)\) associated with the non-split extension of \(O'N\) by a centre of order 3 is universal.
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 51E24
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 51D20
zbMATH DE Number: 1271727
zbMATH Keywords: geometries
zbMATH Keywords: flag-transitive actions
zbMATH Keywords: finite simple groups
zbMATH Keywords: O'Nan sporadic simple group
zbMATH Keywords: maximal parabolics
zbMATH Keywords: triple covers
reviewed by: Alexander A. Ivanov
describes a project that uses: Magma
MaRDI profile type: Publication
full work available at URL: jcta.1998.2912
OpenAlex ID: W1981362271
cites work: The Magma algebra system. I: The user language
cites work: Diagrams for geometries and groups
cites work: Q3041871
cites work: Q3697813
cites work: Q4763522
cites work: Q3172816
cites work: Q4385461
cites work: The residually weakly primitive geometries of \(S _{5} {\times}\)2
cites work: Q3684278
cites work: Classifying geometries with CAYLEY
cites work: The classification of finite simple groups I. Simple groups and local analysis
cites work: Q4385463
cites work: A geometry for the O'Nan-Sims group connected with the Petersen graph
cites work: Q3027217
cites work: Some Evidence for the Existence of a New Simple Group
cites work: The maximal subgroups of the O'Nan group
cites work: Q3688532
DOI: 10.1006/JCTA.1998.2912
Recommended article: On the geometry of {\textsf{O'N}}
Recommended article: An alternative existence proof of the geometry of Ivanov-Shpectorov for O'Nan's sporadic group
Recommended article: A rank 3 geometry for the O'Nan group connected with the Livingstone graph
Recommended article: Geometry of sporadic groups. II: Representations and amalgams
Recommended article: A Presentation for J 4
Recommended article: Q4550664
Recommended article: Two rank six geometries for the Higman--Sims sporadic group
Recommended article: Q4535093
Recommended article: The flag-transitive tilde and Petersen-type geometries are all known
Recommended article: The Residually Weakly Primitive Geometries ofJ3
description: scientific article; zbMATH DE number 1271727