\title
{Freyd's generating hypothesis for the stable module category of a $p$-group}

\author{David J. Benson}
\address{Department of Mathematical Sciences\\
University of Aberdeen \\
Meston Building \\
\break King's college, Aberdeen AB24 3UE, Scotland, UK}

\author{Sunil K. Chebolu}
\address{Department of Mathematics \\
University of Western Ontario \\
London, ON N6A 5B7, Canada}

\author{J. Daniel Christensen}
\address{Department of Mathematics \\
University of Western Ontario \\
London, ON N6A 5B7, Canada}

\author{J\'{a}n Min\'{a}\v{c}}
\address{Department of Mathematics\\
University of Western Ontario\\
London, ON N6A 5B7, Canada}

\keywords{Generating hypothesis, stable module category, ghost map}
\subjclass[2000]{Primary 20C20, 20J06; Secondary 55P42}

\begin{abstract}
Freyd's generating hypothesis, interpreted in the stable module 
category of a finite $p$-group $G$, is the statement that
a map between finite-dimensional $kG$-modules factors through a projective if
the induced map on Tate cohomology is trivial. We show that Freyd's generating
hypothesis holds for a non-trivial $p$-group $G$ if and only if $G$ is either
$\mathbb{Z}/2$ or $\mathbb{Z}/3$. We also give various conditions which are
equivalent to the generating hypothesis.
\end{abstract}