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

where the edge in the middle indicates the geometry of vertices and edges o f the Petersen graph. The elements corresponding to the nodes from the left to the right on the diagram P-3(3) are called points, lines, triangles and planes, respectively. The residue in G of a point is the P-3-geometry G(Ma t(22)) of the Mathieu group of degree 22 and the residue of a plane is the P-3-geometry G(Alt(7)) of the alternating group of degree 7. The geometries G(Mat(22)) and G(Alt(7)) possess 3-fold covers Q(3Mat(22)) and G(3Alt(7)) which are known to be universal. In this paper we show that G is simply con nected and construct a geometry (G) over tilde which possesses a 2-covering onto G. The automorphism group of (G) over tilde is of the form 3(23)MCL; the residues of a point and a plane are isomorphic to G(3Mat(22)) and G(3Al t(7)), respectively. Moreover, we reduce the classification problem of all flag-transitive P-n(m)-geometries with n, m greater than or equal to 3 to t he calculation of the universal cover of (G) over tilde.