Groups of projective planes with differing numbers of point and line orbits (Q397913)

It is known that a group of collineations of a finite projective plane has the same number of orbits on the set of points and on the set of lines. The question was posed whether this is also true for every group of collineations of an infinite projective plane. The authors give in this article examples of infinite projective planes with collineation groups having different numbers of orbits on the set of points and on the set of lines. Some of their examples are Desarguesian projective planes.NEWLINENEWLINEThey yield collineation groups of Desarguesian planes with two point orbits and at least three line orbits, but their method can also be used to obtain other numbers of point and line orbits. The authors also construct collineation groups of projective planes with arbitrary numbers of point and line orbits.