Remarks on the Prime Power Conjecture for Projective Planes

Dieter Jungnickel
University of Augsburg, Germany
(joint work with Aart Blokhuis and Bernhard Schmidt)

Projective planes of order $n$ have been constructed for all prime powers $n$, but for no other values of $n$. The prime power conjecture (PPC) asserts that a projective plane of order $n$ exists if and only if $n$ is a prime power. The nonexistence of a projective plane of order $n$ is known for all $n\equiv 1,2 \bmod 4$ which are not sums of two squares (Bruck-Ryser theorem), for $n=10$ (Lam et al. 1989) and for no other values of $n$.

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

Theorem 1   Let $G$ be an abelian collineation group of order $n^2$ of a projective plane of order $n$. Then $n$ is a prime power, say $n=p^b$. If $p>2$, then $G$ has rank at least $b+1$.

As a consequence, one gets an interesting corollary for planar functions. Let $H$ and $K$ be groups of order $n$. A planar function of degree $n$ is a map $f:H\to K$ such that the induced map $f_h:x\mapsto f(hx)f(x)^{-1}$ is bijective for every $h\in H\setminus \{1\}$. Any planar function from $H$ to $K$ gives rise to a projective plane of order $n$ on wich $H\times K$ acts as a collineation group. Hence one has

Corollary 2   If there is a planar function of degree $n$ between abelian groups, then $n$ is a prime power.

For background, see:

A. Pott: Finite geometry and character theory. Lecture Notes in Mathematics 1601, Springer 1995.