Some rank five geometries related to the Mathieu group \(M_{23}\) (Q5947368)

Only few geometries are known on which the Mathieu group \(M_{23}\) acts flag-transitively. One such rank five diagram geometry was constructed by \textit{A. Pasini} [Eur. J. Comb. 17, 657-671 (1996; Zbl 0860.51005)]. NEWLINENEWLINENEWLINEIn the paper under review the author constructs six rank five diagram geometries on which \(M_{23}\) acts flag-transitively and residually weakly primitively. The first geometry is the one given by Pasini for which the author gives a different construction. He uses the representation of \(M_{23}\) as the automorphism group of the Steiner system \(S(4,7,23)\). Elements of type 1, 2, 3, 4, 5 are the points, pairs of points, triples of points, quadruples of points, and the blocks of size 8 of \(S(4,7,23)\), respectively. Incidence is inclusion. NEWLINENEWLINENEWLINETwo further geometries are given, both based on \(S(4,7,23)\) where the elements of various types are formed from the points (this set is repeated), pairs of points, triples of points, quadruples of points, the blocks of size 7, or the blocks of size 8 of \(S(4,7,23)\) and incidence is not necessarily inclusion. The automorphism group of each of these three geometries is \(M_{23}\). NEWLINENEWLINENEWLINEApplying the process of doubling the author obtains from these three geometries three further firm and residually connected rank five geometries on which \(M_{23}\) acts flag-transitively.