On the maximal rank of primitive residually connected geometries for \(M_{22}\)
type: Item
instance of: scholarly article
title: On the maximal rank of primitive residually connected geometries for \(M_{22}\)
zbMATH Open document ID: 1271.51003
author: Nayil Kilic
author: Peter J. Rowley
published in: Journal of Algebra
publication date: 2013-02-21T00:00:00Z
review text: The authors show that there are no flag transitive residual connected and primitive geometries \(\Gamma\) for \(G= M_{22}\) of rank of least 7. In fact, there is one of rank 6. Let \(n\) be the rank of \(\Gamma\) and \(F\) be a flag of rank \(n\). Identify the flag with the set \(\{1,\dots,n\}\). Then the assumptions about \(\Gamma\) yield that for all \(I,J\subseteq\{1,\dots,n\}\) with \(I\subseteq J\) and \(|J\setminus I|= 1\) we have that \(G_J\) is maximal in \(G_I\). In particular, \(G_i\) is a maximal subgroup of \(G\) for all \(i\in\{1,\dots,n\}\). Hence the proof is a careful study of the maximal subgroups of \(G\) and maximal subgroups of various intersections.
reviewed by: Gernot Stroth
Mathematics Subject Classification ID: 51E10
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 6138247
zbMATH Keywords: Mathieu groups
zbMATH Keywords: Steiner system
zbMATH Keywords: group geometries
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2012.03.034
OpenAlex ID: W2027542585
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DOI: 10.1016/J.JALGEBRA.2012.03.034
description: scientific article; zbMATH DE number 6138247