Finite geometries of typeC3with flag-transitive groups

✍ Scribed by Michael Aschbacher

• Publisher: Springer
• Year: 1984
• Tongue: English
• Weight: 246 KB
• Volume: 16
• Category: Article
• ISSN: 0046-5755
• DOI: 10.1007/bf00146830

✦ Synopsis

Recently there has been renewed interest in a class of geometries introduced by Tits many years ago. Part of this interest stems from Tits' paper [-6] which characterizes buildings as the simply connected geometries with Coxeter diagram in which all residues of type C 3 and H 3 are buildings.

Define a finite rank 2 geometry to be classical if it is a generalized n-gon defined by the cosets of parabolic subgroups of a finite group of Lie type and lie rank 2. Let ~ be the class of geometries F admitting a flag-transitive group of automorphisms G such that each rank 2 residue of F is either classical or a generalized 2-gon, and such that the stabilizer in G of each proper residue is finite, cg is of interest in the study of finite groups.

As there are no finite groups of Lie type with diagram H2, there are no geometries in f# with residues of type H 3. The following result clears up in part the remaining ambiguity involving residues of type C 3 . THEOREM 1. Let F be a geometry in fg of type C 3 and G a flag-transitive group of automorphisms ofF. Then either F is a polarspace or G ~-A 7 and F is the 7-geometry.