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On k-nets of order \((k-1)^ 2\) admitting improper collineations. (English) Zbl 0558.51006
A permutation of a finite net is a collineation if the images of two points joined by a line is always a pair of points joined by a line. If the image of each line is a line, the collineation is proper, otherwise improper.
The author studies nets with k parallel classes and order \((k-1)^ 2\) (i.e., if the order is n, the number of parallel classes is \(1+\sqrt{n})\). He postulates the existence of an improper collineation and then shows that the net is what the reviewer calls a derivable net. He shows that the net can be embedded in a Desarguesian plane of order \((k-1)^ 2\) with slopes taken from \(\infty U\) GF(k-1).
Reviewer: T.Ostrom
MSC:
51A45 Incidence structures embeddable into projective geometries
51A15 Linear incidence geometric structures with parallelism
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