Geometry Seminar


On Quadratic Extensions of Cyclic Projective Planes


Mr. Hiu Fai Law

The University of Hong Kong




A projective plane consists of a set of points and lines such that two lines intersect at a unique point and two points lie on a unique line. It is known that a projective plane is isomorphic to P2(R) for some division ring R if and only if it is Desarguesian, but that there are projective planes that are non-Desarguesian. When the set of points and the set of lines are finite, R has to be a finite field so that the order of a finite projective Desarguesian plane must be a power of a prime. The most well-known unsolved problem in this field is to see whether the order of any finite projective plane is a power of a prime.

A projective plane admitting a cyclic collineation group is called cyclic. The study of cyclic projective plane is closely related to that of cyclic planar difference sets. Singer proves that every Desarguesian plane is cyclic. Partial results on the converse have been given by Bruck by showing that cyclic planar difference sets of certain square orders are unique, hence cyclic projective planes of those orders are Desarguesian. We will investigate his methods and interpret them from a more geometric viewpoint.



April 6, 2006 (Thursday)


4:00 – 5:00pm


Room 517, Meng Wah Complex, HKU



All are welcome