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\title{The classification of endo-trivial modules}
\author[Jon Carlson]{Jon F. Carlson$^*$}
\thanks{* Partly supported by a grant from NSF}
\address{J. F. C.: Department of Mathematics, University of Georgia,
Athens, Georgia 30602, USA}
\author[Jacques Th\'evenaz]{Jacques Th\'evenaz}
\address{J. T.: Institut de g\'eom\'etrie, alg\`ebre et topologie,
EPFL, CH-1015 Lausanne, Switzerland}
\date\today
\begin{document}
\maketitle
\begin{abstract}
This paper provides the final step for the
classification of all endo-trivial $kP$-modules, where $P$ is a finite
$p$-group and $k$ is a field of characteristic~$p$. The
classification had been known for a
long time if $P$ has a maximal elementary abelian subgroup of rank~1,
i.e. if $P$ cyclic or generalized quaternion.

Let $T(P)$ be the group of equivalence classes of endo-trivial
$kP$-modules. The classification of endo-trivial $kP$-modules is
equivalent to the complete description of the structure of~$T(P)$. It
was proved by the authors in an earlier paper
that $T(P)$ is free abelian if $P$ is not
cyclic, quaternion or semi-dihedral, and that $T(P)\cong {\Bbb Z}$ in
most
cases, namely when every maximal elementary abelian subgroup of~$P$ has
rank at least~3. The torsion-free rank
of $T(P)$ was described in group-theoretic terms by
Alperin. The chief contribution of this paper is to
show that, outside of the exceptional cases mentioned above,
the relative syzygies found by Alperin form a free basis for
$T(P)$ and do not just generate a subgroup of finite index.
This is the final
step in the classification of the elements of $T(P)$.
\end{abstract}
\end{document}