Two rank six geometries for the Higman--Sims sporadic group
It is a rather unusual phenomenon that a sporadic simple group acts flag transitively on a ``nice'' geometry of high rank (``nice'' in the sense of satisfying both some general and specific axioms, in particular non-degeneracy axioms). In the present paper, the author constructs two such geometries for the sporadic Higman-Sims group \(HS\). The geometries have both rank 6, which is rather large. The geometries have been discovered by computer, but the author gives a computer free construction, starting with the action of the alternating group \(A_5\) on the Petersen graph, and extending to rank 3, 4 and 5 geometries for the groups \(A_6\), \(L_3(4)\) and \(M_{22}\), respectively. At last, an extension to \(HS\) is defined, and the author obtains his first geometry of rank 6. The other geometry is a distortion of the first one by changing the incidence relation slightly. The residues with connected diagram provide new rank 5 geometries for Mathieu's group. \(M_{22}\). The diagrams of the new geometries are stars and thin in all the endnodes. The rank 2 residues are complete or Petersen graphs.
scientific article; zbMATH DE number 2182135