Groups for geometries in given diagrams. II: On a characterization of the group Aut(HS)
type: Item
instance of: scholarly article
title: Groups for geometries in given diagrams. II: On a characterization of the group Aut(HS)
zbMATH Open document ID: 0716.20001
DOI: 10.1016/S0195-6698(13)80081-9
author: Antonio Pasini
published in: European Journal of Combinatorics
publication date: 1991-01-01T00:00:00Z
review text: The purpose of the present paper is to prove that the c.C\({}_ 2\) geometry \(\Gamma\) belonging to Aut(HiS) discovered by \textit{S. Yoshiara} [Eur. J. Comb. 11, 81-93 (1990; Zbl 0697.51008)] is simply connected. This has been shown before by \textit{R. Weiss} and \textit{S. Yoshiara} [J. Algebra 133, 182-196 (1990; Zbl 0704.20015)] using the computer system CAYLEY. The proof given in the paper under review does not use a computer at all. The proof is based on the theory of coverings established by \textit{M. Ronan} [Eur. J. Comb. 1, 259-269 (1980; Zbl 0553.51004); Finite Geometries and Designs, Lond. Math. Soc. Lect. Note Ser. 49, 316-331 (1981; Zbl 0467.51015)]. The basic idea is to show that in the collinearity graph for \(\Gamma\) any closed path can be split into triangles. From this it easily follows that the group of decktransformations of \(\Gamma\) has to be trivial, i.e. \(\Gamma\) is simply connected.
reviewed by: Gernot Stroth
Mathematics Subject Classification ID: 20B25
Mathematics Subject Classification ID: 51E30
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 51E25
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20D45
zbMATH DE Number: 4179634
zbMATH Keywords: Higman-Sims-group
zbMATH Keywords: \(c.C_ 2\) geometry
zbMATH Keywords: coverings
zbMATH Keywords: group of decktransformations
zbMATH Keywords: simply connected
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4179634