Constructions of rank five geometries for the Mathieu group \(M_{22}\)
type: Item
instance of: scholarly article
title: Constructions of rank five geometries for the Mathieu group \(M_{22}\)
zbMATH Open document ID: 1057.51006
author: Dimitri Leemans
published in: Journal of Geometry
publication date: 2004-09-22T00:00:00Z
review text: Most known geometries on which the Mathieu group \(M_{22}\) acts flag-transitively have a linear diagram except for a construction by \textit{T. Meixner} and \textit{A. Pasini} [Atti Semin. Mat. Fis. Univ. Modena 44, No.~1, 67--80 (1996; Zbl 0866.51006)]. In the paper under review the author gives constructions for nine rank five geometries for \(M_{22}\), none of which has a linear diagram. All nine geometries are based on the Steiner system \(S(3,6,22)\) contained in the Steiner system \(S(5,8,24)\) associated with the Mathieu groups \(M_{22}\) and \(M_{24}\), respectively. The first geometry \(\Gamma_1\) is the Meixner-Pasini geometry and the next three are somewhat related to \(\Gamma_1\). \(\Gamma_2\) is obtained from \(\Gamma_1\) by doubling; in \(\Gamma_3\) a construction introduced by the author [J. Comb. Theory Ser. A 95, 365--372 (2001; Zbl 0990.51005)] for \(M_{23}\) is applied and triples of points are replaced by points of \(\Gamma_1\); \(\Gamma_4\) is obtained from \(\Gamma_3\) by doubling. The other five rank five geometries for \(M_{22}\) are new. In \(\Gamma_3\) triples of points are replaced by octads. The remaining four geometries for \(M_{22}\) use some special partitions of the point set of \(S(3,6,22)\) such that the group \(M_{22}\) acts transitively on the set of these partitions. Each of these four geometries has a star diagram and some have a rank four truncation in common. In particular, the geometries \(\Gamma_7\), \(\Gamma_8\) and \(\Gamma_9\) are obtained from rank four trunctions of \(\Gamma_6\) of type \(\{1,3,4,5\}\), \(\{1,2,3,5\}\) and \(\{1,2,4,5\}\), respectively. Only the descriptions of the geometries are given, that is, definitions for elements of types one to five and incidence, and their diagrams. The verification of the incidence axioms and that the geometries are firm, residually connected and having \(M_{22}\) acting flag-transitively on them is left to the reader.
reviewed by: Günter F. Steinke
Mathematics Subject Classification ID: 51E24
Mathematics Subject Classification ID: 20D08
zbMATH DE Number: 2101783
zbMATH Keywords: Mathieu group
zbMATH Keywords: sporadic group
zbMATH Keywords: incidence geometry
zbMATH Keywords: Steiner system
MaRDI profile type: Publication
full work available at URL: s00022-003-1736-5
OpenAlex ID: W1977129754
DOI: 10.1007/S00022-003-1736-5
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description: scientific article; zbMATH DE number 2101783