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{\large {\bf Remarks on the Prime Power Conjecture for Projective Planes}}
\\ [+.1in]
{\bf Dieter Jungnickel\\
University of Augsburg, Germany} \\[2mm]
(joint work with Aart Blokhuis and Bernhard Schmidt)
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\noindent
\vspace{2mm}

Projective planes of order $n$ have been constructed for all prime powers $n$,
but for no other values of $n$. The {\it 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

\begin{thm} \label{ppc1}
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 $b+1$.
\end{thm}

As a consequence, one gets an interesting corollary for planar functions.
Let $H$ and $K$ be groups of order $n$.
A {\it 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

\begin{cor} \label{planar}
If there is a planar function of degree $n$ between abelian groups, then
$n$ is a prime power.
\end{cor}

\vspace{1mm}

For background, see:

\vspace{2mm}

A. Pott:
{\it Finite geometry and character theory.}
  Lecture Notes in Mathematics \textbf{1601}, Springer 1995.

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