Math 5392 Finite Geometries I Summer 2004
Dr. Cordero
Projective Planes
Additional Problems
- A statement is self-dual if it is the same as its dual. Give an example of a self-dual statement.
- Prove that there is no projective plane on 29 points.
- Show that a finite projective plane always has an odd number of points.
- Draw a projective plane of order 4. (Hard: give the line sets.)
- Find the possible orders of subplanes which may be contained in any projective plane of order 39.
- Let Π be the Fano plane. Show that the completion of any quadrangle of Π is always Π.
- Find a central collineation of the projective plane of order 3.
- Find two non-identity
collineations of the projective plane of order 3, for fixed 
and 
. - A correlation of a projective plane is a 1-1 map from the set of all points and lines onto itself, such that a point is mapped to a line and a line to a point, and such that incidence is preserved (i.e.
if and only if 
). Find a correlation of the projective plane of order 2. - Show that the projective planes of orders 2 and 3 have complete sets of central collineations.