# zbMATH — the first resource for mathematics

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
##### Keywords:
net; proper collineation; improper collineation
Full Text: