5 - Fischer spaces

Summary

Fischer classified the groups generated by a conjugacy class 𝐷 of 3-transpositions (involutions such that the product of any two has order at most 3) and discovered three new sporadic groups that bear his name. These groups are rank 3 groups: 𝐷 carries in a natural way the structure of a geometry with lines of length 3 and the structure of a rank 3 graph. We review some properties of these geometries, called Fischer spaces, and mention Fischer’s group-theoretic classification. We discuss the examples and give detailed parameter information. We briefly discuss cotriangular spaces and Shult’s classification, and Hall’s classification of copolar spaces. Finally we classify with full proofs all lax embeddings of (finite) symplectic copolar spaces in projective space. We use some specific such embedded copolar spaces to construct two rank 3 graphs of affine type, one with 2401 vertices, the other with 6561 vertices.