Ein gruppentheoretischer Aufbau der äquiformen Geometrie
The author studies the group of similarities of a euclidean, a Galilean, and a Minkowskian plane, i.e. an affine plane over a field K of characteristic \(\neq 2\), together with an orthogonality relation defined via a symmetric bilinear form \(((x_ 1,x_ 2),(y_ 1,y_ 2))\mapsto x_ 1y_ 1+kx_ 2y_ 2\) with some \(k\in K\) (-k not a square, \(k=0\), - k a square in \(K\setminus \{0\}\), respectively, gives a euclidean, a Galilean, a Minkowskian plane, respectively). A system of seven axioms describes these groups where the distinct types of planes are distinguished by two further axioms and their negations. With these two axioms the author obtains a euclidean Bachmann group, a parabolic Hjelmslev group, or a Minkowskian group, respectively. To each of these groups well known representation theorems apply [see \textit{F. Bachmann}, ''Aufbau der Geometrie aus dem Spiegelungsbegriff'' (1973; Zbl 0254.50001), \textit{H. Struve}, ''Ein spiegelungsgeometrischer Aufbau der Galileischen Geometrie'', to appear in the same journal, and \textit{H. Wolff}, Math. Ann. 171, 144-193 (1967; Zbl 0148.146), respectively]. Since the group of similarities is not generated by reflections, some additional work is required.
scientific article; zbMATH DE number 3901537