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4-transitivity and 6-figures in some Moufang-Klingenberg planes. (English) Zbl 1131.51003

It is proved that the collineation group of the Moufang-Klingenberg plane over the local alternative ring \(R=A+A\varepsilon\) with \(A\) an alternative division ring and \(\varepsilon^2=0\), is transitive on quadrangles. As a corollary the coordinatizing local alternative ring of such a plane is independent of the choice of the quadrangle used in the coordinatization. Also the concept of \(6\)-figures is extended from ordinary projective planes to these Moufang-Klingenberg planes and it is shown that any \(6\)-figure corresponds to a unique invertible \(m\) in \(R\).

MSC:

51C05 Ring geometry (Hjelmslev, Barbilian, etc.)
51A35 Non-Desarguesian affine and projective planes
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