On Quadratic Extensions of Cyclic Projective Planes
Mr. Hiu Fai Law
Abstract
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.
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Room 517, Meng Wah Complex, HKU |
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All are welcome |
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