On the uniqueness of the \(Co_ 1\) 2-local geometry
type: Item
instance of: scholarly article
title: On the uniqueness of the \(Co_ 1\) 2-local geometry
zbMATH Open document ID: 0645.51014
DOI: 10.1007/BF00191926
author: Yoav Segev
published in: Geometriae Dedicata
publication date: 1988-01-01T00:00:00Z
review text: In this remarkable paper the author shows the uniqueness of a certain geometry for \(Co_ 1\) discovered by \textit{M. A. Ronan} and \textit{S. D. Smith} [Proc. Symp. Pure Math. 37, 283-289 (1980; Zbl 0478.20015)]. This is a geometry consisting of points, lines, planes and hyperplanes such that the residue of a point is a \(M_{24}\)- geometry, also discovered by M. Ronan and S. Smith, and the residue of a hyperplane consists of the points, lines and one class of maximal singular subspaces of the \(0\) \(+_ 8(2)\)- building. The author shows that a geometry \(\Gamma\) having this kind of residue structure and possessing a flag transitive automorphism group G, is uniquely determined and \(G\cong Co_ 1\). The proof of this result is divided into two parts. First the stabilizers of a point and a hyperplane are determined. Then it is shown that G admits a rank 6 representation on the points. Using this information it is possible to show that the collinearity graph of \(\Gamma\) is uniquely determined.
Mathematics Subject Classification ID: 51E30
Mathematics Subject Classification ID: 20D08
zbMATH DE Number: 4052443
zbMATH Keywords: first Conway group
zbMATH Keywords: uniqueness
zbMATH Keywords: residue
zbMATH Keywords: flag transitive automorphism group
reviewed by: Gernot Stroth
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00191926
OpenAlex ID: W4211222346
description: scientific article; zbMATH DE number 4052443