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A characterization of the Petersen-type geometry of the McLaughlin group

Published online by Cambridge University Press:  01 January 2000

B. BAUMEISTER
Affiliation:
Fachbereich Mathematik und Informatik, Institut für Algebra und Geometrie, Martin-Luther-Universität Halle-Wittenberg, D-06099 Halle (Saale), Germany
A. A. IVANOV
Affiliation:
Department of Mathematics, Imperial College, 180 Queen's Gate, London SW7 2BZ, England
D. V. PASECHNIK
Affiliation:
Department of Computer Science, Utrecht University P.O. Box 80089 3508 TB Utrecht, The Netherlands

Abstract

The McLaughlin sporadic simple group McL is the flag-transitive automorphism group of a Petersen-type geometry [Gscr ] = [Gscr ](McL) with the diagram

diagram here

where the edge in the middle indicates the geometry of vertices and edges of the Petersen graph. The elements corresponding to the nodes from the left to the right on the diagram P33 are called points, lines, triangles and planes, respectively. The residue in [Gscr ] of a point is the P3-geometry [Gscr ](Mat22) of the Mathieu group of degree 22 and the residue of a plane is the P3-geometry [Gscr ](Alt7) of the alternating group of degree 7. The geometries [Gscr ](Mat22) and [Gscr ](Alt7) possess 3-fold covers [Gscr ](3Mat22) and [Gscr ](3Alt7) which are known to be universal. In this paper we show that [Gscr ] is simply connected and construct a geometry [Gscr ]˜ which possesses a 2-covering onto [Gscr ]. The automorphism group of [Gscr ]˜ is of the form 323McL; the residues of a point and a plane are isomorphic to [Gscr ](3Mat22) and [Gscr ](3Alt7), respectively. Moreover, we reduce the classification problem of all flag-transitive Pmn-geometries with n, m [ges ] 3 to the calculation of the universal cover of [Gscr ]˜.

Type
Research Article
Copyright
The Cambridge Philosophical Society 2000

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