Groups for geometries in given diagrams. II: On a characterization of the group Aut(HS)
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.
scientific article; zbMATH DE number 4179634
scientific article