Flag-transitive Buekenhout geometries

This chapter discusses the concept of flag-transitive Buekenhout geometries. More than one half of sporadic simple groups are known to act flag-transitively on finite geometries belonging to Buekenhout diagrams obtained from Coxeter diagrams replacing some of strokes for projective planes with strokes for circular spaces or for dual circular spaces. The chapter recalls that a circular space is a finite linear space with lines of size 2—namely, the system of vertices and edges of a complete graph. Classification theorems exist for some classes of geometries; the chapter surveys some of those theorems, choosing certain diagrams for which a classification is known or at least substantial progresses have been done in that direction. The chapter also discusses some background on Lie diagrams, the Neumaier geometry, spherical and nonspherical diagram, and affine constructions. © 1992, Elsevier B.V. All rights reserved.