Hall, J. I.; Shpectorov, S. V. \(P\)-geometries of rank 3. (English) Zbl 0971.51010 Geom. Dedicata 82, No. 1-3, 139-169 (2000). A rank three \(P\)-geometry is a geometry with points, lines and planes, such that the point-residue is the dual of the Petersen graph, the line-residue is a generalized digon and the plane-residue is a projective plane of order three. The authors show that if \(\Gamma\) is a rank three \(P\)-geometry such that any two lines intersect in at most one point and any three pairwise collinear points belong to a plane, then \(\Gamma\) is either the 2-local geometry of \(M_{22}\) or the geometry of \(3\cdot M_{22}\). Reviewer: Dimitri Leemans (Bruxelles) Cited in 1 ReviewCited in 2 Documents MSC: 51E24 Buildings and the geometry of diagrams 20D08 Simple groups: sporadic groups 05B25 Combinatorial aspects of finite geometries Keywords:diagram geometry; \(P\)-geometries; sporadic groups PDFBibTeX XMLCite \textit{J. I. Hall} and \textit{S. V. Shpectorov}, Geom. Dedicata 82, No. 1--3, 139--169 (2000; Zbl 0971.51010) Full Text: DOI