Ank 3 Characterizations of Classica

Ever since Higman’s ground-breaking work $1, there have been many rank 3 characterizations of the classical symplectic, unitary and orthogonal groups and geometries. The purpose of this paper is to add another such characterization to that pile. Take a finite symplectic, unitary or orthogonal geometry having a totally singular (projective) plane. Let R be the number of singular points # x orthogonal to a given singular point X; let I be the number of singular points not orthogonal to x. Finally, let G be a rank 3 permutation gr-oup such that the stabilizer al’ a point has orbits of lengths 1, k, and 1. We will show that G can be regarded as an automorpbism group of the given geometry acting on the singular points. This result will be deduced from general theorems which even give some information when singular lines but no singular planes exist. The main idea of the proof is very elementary, essentially the same as that of Rantor [6]* The actual identification of the geometries is made using the theorem of Buekenhout-Shult [I]. We refer to Higman [2] for the relevant background concerning rank 3 groups, and to Higman-McLaughlin [5], Perin [7], and Stark [B] for what is known about rank 3 subgroups of symplectic, unitary, and orthogonal groups. The precise statements are as follows.