\title{On saturated fusion systems and Brauer indecomposability of  Scott modules}
\author{Radha Kessar, Naoko Kunugi, Naofumi Mitsuhashi}

\address{Radha Kessar\\
Institute of Mathematics\\
University  of Aberdeen \\
Fraser Noble Building \\
Aberdeen AB24 3UE\\
U.K.}
\address{Naoko Kunugi\\
Department of Mathematics\\
Tokyo University of Science\\
1-3 Kagurazaka\\
Shinjuku-ku\\
Tokyo 162-8601\\
Japan}
\address{Naofumi Mitsuhashi\\ 
Department of Mathematics\\
Tokyo University of Science\\
1-3 Kagurazaka\\
Shinjuku-ku\\
Tokyo 162-8601\\
Japan}
\maketitle


\begin{abstract}
Let $p$ be a prime number, $G$  a finite group,
$P$ a $p$-subgroup of $G$ and  $k$   an algebraically closed  field of characteristic $p$.  
We  study the   relationship between  the  category $\Ff_P(G)$ and  
the behavior  of  
$p$-permutation $kG$-modules  with vertex $P$ under the Brauer construction.  
We give  a  sufficient condition  for   $\Ff_P(G)$ to be a saturated 
fusion system.   We   prove that   for Scott modules with   abelian vertex,  our condition is
also necessary.  In order to obtain our results,  we prove a criterion  for the categories arising from the 
data  of $(b, G)$-Brauer pairs   in the sense of  Alperin-Brou\'e and 
Brou\'e-Puig  to  be saturated fusion systems  on the underlying 
$p$-group. 
\end{abstract}