Projective planes of order have been constructed for all prime powers
,
but for no other values of
. The prime power conjecture
(PPC) asserts that a
projective plane of order
exists if and only if
is a prime power.
The nonexistence of a projective plane of order
is known for all
which are not sums of two squares (Bruck-Ryser theorem), for
(Lam et al. 1989) and for no other values of
.
The present methods of Mathematics seem to offer no hope of settling the PPC. Thus one adds extra assumptions, usually requiring the existence of a particularly nice type of collineation group, in particular with a reasonably large abelian group (the extreme case being that of a Singer group). We will survey previous results in this area and settle one of the open cases by the following
As a consequence, one gets an interesting corollary for planar functions.
Let and
be groups of order
.
A planar function of degree
is a map
such that
the induced map
is bijective for every
.
Any planar function from
to
gives rise to a projective plane of order
on wich
acts as a collineation group. Hence one has
For background, see:
A. Pott: Finite geometry and character theory. Lecture Notes in Mathematics 1601, Springer 1995.