Abstract
Leißner [2] proved that the class of all incidence structures with similarity-relation coinzides with, the class of the algebraically defined geometries [F,T], where F denotes a neardomain over a subdomain T.
In this paper we characterize those geometries, where F is a near-resp. (skew-) field by additional similarity axioms. At first we show that a subdomain T of a neardomain F is itself a neardomain iff−1εT and characterize this fact geometrically. As a consequence every subdomain of a near-resp.(skew-) field has to be a near-resp. (skew-) field too. In §4 we get as a corollary that projective planes admit no sharply twice transitive groups of collineations.
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Literatur
Kerby, W.; Wefelscheid, H.: Bemerkungen über Fastbereiche und scharf zweifach transitive Gruppen, Abh. Math. Sem. Univ. Hamburg 37 (1971) 23–32.
Leißner, W.: Ein axiomatischer Aufbau der Ähnlichkeitsgeometrie, J. Geometrie 5 (1974) 117–146.
Leißner, W.: Ein Stufenaufbau der Fastbereiche, Fastkörper und Körper aus ihrer multiplikativen Gruppe, Abh. Math. Sem. Univ. Hamburg 46 (1977) 57–91.
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Brockhaus, E. Zum Stufenaufbau der Inzidenzstrukturen mit Ähnlichkeitsrelation. J Geom 10, 106–125 (1977). https://doi.org/10.1007/BF01933065
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DOI: https://doi.org/10.1007/BF01933065