Projective extensions of Kirkman systems as substructures of projective planes
In a Kirkman system, each line contains the same number k of points. The lines occur in parallel classes so that each point is exactly on one line of each parallel class. Just as in the special case of an affine plane, we can adjoin one new point for each parallel class and one new line which contains all of the new points. This is the ``projective extension'' of the Kirkman system or the ``extended Kirkman system''. This paper deals with the case where an extended Kirkman system is embedded in a projective plane. It is not known whether or not there is a projective plane of order 15; there is a Kirkman system with 15 points and \(k=3\). A natural question is whether it might be possible to use the (extended) Kirkman system to construct a projective plane of order 15 with the extended Kirkman system embedded in the plane. The authors show that if a plane of order 15 contains an extended Kirkman system and admits a collineation \(\sigma\) of order 7 which fixes the Kirkman system, then the full collineation group of the plane is equal to \(<\sigma >\).
scientific article; zbMATH DE number 4054452
scientific article