On a rank five geometry of Meixner for the Mathieu group \(M_{12}\) (Q5946252)

The author proves some properties of a coset geometry for \(2 \cdot M_{12}\) discovered by \textit{T. Meixner} [Atti Sem. Mat. Fis. Univ. Modena 44, No. 1, 209-227 (1996; Zbl 1002.51501)]. This geometry extends the \(c. c^*\) geometry obtained from the biplane related to \(L_1(11)\) (by considering the points, pairs of points, the blocks and the natural incidence) to the left and the right with a projective plane of order 1. It can also be seen as an extension of the geometry of singletons, pairs and triples of a 6-set by the dual of this geometry. The properties proved by the author in the paper under review include RWPRI (i.e., the automorphism group \(G_\Gamma\) induced in every residue \(\Gamma\) acts flag transitively on \(\Gamma\) and there is an element \(x\) in every residue \(\Gamma\) whose stabilizer \((G_\Gamma)_x\) is maximal in \(G_\Gamma)\) and (IP)\(_2\) (i.e., every rank 2 residue is either a partial linear space or a generalized digon). Apparently, \(M_{12}\) does not admit any rank 6 geometry having these two properties, and so the paper under review proves that the upper bound for the rank of such geometries related to \(M_{12}\) is exactly 5.