\title{The Dade group of a metacyclic $p$-group}
\author{Nadia Mazza\thanks{This work is a part of a doctoral thesis 
in preparation at the University of Lausanne, under the supervision 
of Prof. Jacques Th\'evenaz}\\
Institut de 
Math\'ematiques\\
Universit\'e de Lausanne\\
CH-1015 
Lausanne}


\begin{abstract} The Dade group $D(P)$ of a finite $p$-group $P$, 
formed by equivalence classes of endo-permutation modules, is a 
finitely generated abelian group. Its torsion-free rank equals the 
number of conjugacy classes of non-cyclic subgroups of $P$ and it is 
conjectured that every non-trivial element of its torsion subgroup 
$D^t(P)$ has order $2$, (or also $4$, in case $p=2$). The group 
$D^t(P)$ is closely related to the injectivity of the restriction map 
$\Res:T(P)\rightarrow\prod_E T(E)$ where $E$ runs over elementary 
abelian subgroups of $P$ and $T(P)$ denotes the group of equivalence 
classes of endo-trivial modules, which is still unknown for (almost) 
extra-special groups ($p$ odd). As metacyclic $p$-groups have no 
(almost) extra-special section, we can verify the above conjecture in 
this case. Finally, we compute the whole Dade group of a metacyclic 
$p$-group.
\end{abstract}