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

Abstract

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