The P-geometry for \(M_{23}\) has no non-trivial 2-coverings. (English) Zbl 0712.51009
A P-geometry is an incidence geometry whose Buekenhout-diagram extends the diagram of a projective space of order 2 linearly with a “P- stroke”, “P” denoting the geometry of the Peterson graph. Such geometries are closely related to certain sporadic simple groups (acting flag transitively). So it would be interesting to have a classification of such flat transitive geometries. In the rank 3 case, this was achieved by the second author [Investigations in the Algebraic Theory of Combinatorial Objects, Proc. Semin., Moskva 1985, 112-123 (1985; Zbl 0661.20009)], where two examples arise, both related to \(M_{22}\) and one being the triple cover of the other. In the paper under review, similar techniques are used to attack the rank 4 case (e.g. calculating universal closures of group amalgams). The rank 4 P-geometries fall into 3 classes: a class related to \(M_{23}\), one to \(Co_ 2\) and one to \(J_ 4\). It is shown that the example related to \(M_{23}\) has no proper covers, i.e. it is 2-simply connected. Hence the class \(M_{23}\) only contains this unique example. This shows that a full classification in the rank 4 case seems within reach.
Reviewer: H.Van Maldeghem
Citations:
Zbl 0661.20009References:
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