On flat flag-transitive \(c.c^*\)-geometries
type: Item
instance of: scholarly article
title: On flat flag-transitive \(c.c^*\)-geometries
zbMATH Open document ID: 0872.51009
DOI: 10.1023/A:1008662500378
author: Barbara Baumeister
author: Antonio Pasini
published in: Journal of Algebraic Combinatorics
publication date: 1997-10-15T00:00:00Z
review text: A \(c.c^*\)-geometry is a rank 3 Buekenhout-geometry with points, lines and planes such that the residue of a point (respectively plane) is a complete graph (respectively dual complete graph) and the residue of a line is a digon. Such a geometry is flat if all points are incident with all planes. The paper under review characterizes all flat flag-transitive \(c.c^*\)-geometries. The result is that there is one sporadic example (related to the symmetric or alternating group of degree 6) and an infinite class obtained by gluing two copies of the point-line geometry of an \(n\)-dimensional affine space over \(GF(2)\). There is some canonical way to do so -- as is explained in the paper -- but also non-canonical ways give rise to flag-transitive examples. There is no explicit classification of these, but some more information (examples, properties) is given in the last section of the paper.
Mathematics Subject Classification ID: 51E24
zbMATH DE Number: 987620
zbMATH Keywords: diagram geometry
zbMATH Keywords: flag-transitivity
zbMATH Keywords: semi-biplane
reviewed by: Hendrik Van Maldeghem
MaRDI profile type: Publication
cites work: Q3708174
cites work: Two new sporadic semibiplanes related to \(M_{22}\)
cites work: Q4878674
cites work: Q3722701
cites work: Parallelism in diagram geometry
cites work: On Groups of Degree N AND n −1, and Highly-Symmetric Edge Colourings
cites work: Finite Permutation Groups and Finite Simple Groups
cites work: One-Factorizations of Complete Graphs with a Doubly Transitive Automorphism Group
cites work: Q3684278
cites work: Q3208669
cites work: Q4200361
cites work: Two new semibiplanes
cites work: Q5762903
cites work: Q4204179
cites work: Finite rank 3 geometries with affine planes and dual affine point residues
cites work: The Affine Permutation Groups of Rank Three
cites work: Gluing two affine spaces
cites work: Q4309517
cites work: The universal covers of certain semibiplanes
full work available at URL: a:1008662500378
OpenAlex ID: W146509494
description: scientific article; zbMATH DE number 987620