\begin{center}
{\bf On Endo-trivial modules for $p$-solvable groups}\\
\emph{Gabriel Navarro\\
Departament d'Algebra\\
Facultat de Matematiques\\
Universitat de Valencia\\
46100 Burjassot\\ Valencia\\
SPAIN\\
 \& \\
Geoffrey R. Robinson,\\
Institute of Mathematics\\
University of Aberdeen\\
Aberdeen\\
AB24 3UE\\
SCOTLAND\\
July 2010}
\end{center}


\medskip
\noindent {\bf Introduction:} In this note, we will prove a
conjecture of Carlson, Mazza and Th\'evenaz [1], namely, we will
prove that if $G$ is a finite $p$-nilpotent group which contains a
non-cyclic elementary Abelian $p$-subgroup and  $k$ is an
algebraically closed field of characteristic $p,$ then all simple
endo-trivial $kG$-modules are $1$-dimensional. In fact, we do rather
more: we prove the analogous result directly in the case that $G$ is
$p$-solvable and contains an elementary Abelian $p$-subgroup of order
$p^{2}.$ Carlson, Mazza and Th\'evenaz had reduced the proof of this
result for $p$-solvable $G$ to the $p$-nilpotent case, (and had
proved the result in the solvable case), but our method is somewhat
different. Our proof does require the classification of finite
simple groups. Specifically, we require the well-known fact that the outer automorphism group
of a  finite simple group of order prime to $p$ has cyclic Sylow $p$-subgroups (see, for example,
Theorem 7.1.2 of Gorenstein,Lyons and Solomon,[3]).