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\(P\)-geometries of rank 3. (English) Zbl 0971.51010

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 \(3\cdot M_{22}\).

MSC:

51E24 Buildings and the geometry of diagrams
20D08 Simple groups: sporadic groups
05B25 Combinatorial aspects of finite geometries
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