Constructions of rank five geometries for the Mathieu group \(M_{22}\)
Most known geometries on which the Mathieu group \(M_{22}\) acts flag-transitively have a linear diagram except for a construction by \textit{T. Meixner} and \textit{A. Pasini} [Atti Semin. Mat. Fis. Univ. Modena 44, No.~1, 67--80 (1996; Zbl 0866.51006)]. In the paper under review the author gives constructions for nine rank five geometries for \(M_{22}\), none of which has a linear diagram. All nine geometries are based on the Steiner system \(S(3,6,22)\) contained in the Steiner system \(S(5,8,24)\) associated with the Mathieu groups \(M_{22}\) and \(M_{24}\), respectively. The first geometry \(\Gamma_1\) is the Meixner-Pasini geometry and the next three are somewhat related to \(\Gamma_1\). \(\Gamma_2\) is obtained from \(\Gamma_1\) by doubling; in \(\Gamma_3\) a construction introduced by the author [J. Comb. Theory Ser. A 95, 365--372 (2001; Zbl 0990.51005)] for \(M_{23}\) is applied and triples of points are replaced by points of \(\Gamma_1\); \(\Gamma_4\) is obtained from \(\Gamma_3\) by doubling. The other five rank five geometries for \(M_{22}\) are new. In \(\Gamma_3\) triples of points are replaced by octads. The remaining four geometries for \(M_{22}\) use some special partitions of the point set of \(S(3,6,22)\) such that the group \(M_{22}\) acts transitively on the set of these partitions. Each of these four geometries has a star diagram and some have a rank four truncation in common. In particular, the geometries \(\Gamma_7\), \(\Gamma_8\) and \(\Gamma_9\) are obtained from rank four trunctions of \(\Gamma_6\) of type \(\{1,3,4,5\}\), \(\{1,2,3,5\}\) and \(\{1,2,4,5\}\), respectively. Only the descriptions of the geometries are given, that is, definitions for elements of types one to five and incidence, and their diagrams. The verification of the incidence axioms and that the geometries are firm, residually connected and having \(M_{22}\) acting flag-transitively on them is left to the reader.
scientific article; zbMATH DE number 2101783