G. R. Robinson,

Amalgams, blocks, weights, fusion systems and finite simple groups

\noindent {\bf ABSTRACT:} We associate an iterated amalgam
of finite groups to a certain class of fusion systems on  
finite $p$-groups ($p$ a prime), in such a way that the $p$-group
of the fusion system is a maximal finite $p$-subgroup of 
the resulting group, unique up to conjugacy, and, furthermore,
the conjugation action of the resulting (usually infinite) group on 
$p$-subgroups induces the original fusion system on the $p$-group.
\footnote{ Since this paper was written, we have been informed that 
I. Leary and R. Stancu are currently writing a paper with 
a different construction to realise a fusion system
on a finite $p$-group via a group. Their work and ours 
are independent of each other}. 
In view of earlier work of Puig and of Broto,Castellana, Grodal, 
Levi and Oliver,  [5], the fusion systems we deal with include 
all saturated fusion systems.


\medskip
The resulting amalgam has free normal subgroups of finite index,
and we examine the images of the group by its maximal free normal
subgroups of finite index; these images all contain (isomorphic copies of)
the original $p$-group (and are generated by the (images of the)
finite groups used in the amalgamation). If there is no non-trivial 
normal $p$-subgroup of the fusion system (equivalently, if the
iterated amalgam constructed has no non-trivial normal 
$p$-subgroup), then the generalized Fitting subgroup of each of these  
homomorphic images is a direct product of non-Abelian simple
groups, each of order divisible by $p$ (and if there is no proper 
non-trivial strongly closed $p$-subgroup in the fusion system, then the
generalized Fitting subgroup of any of these finite groups is 
characteristically simple). 

\medskip
We note that the (finite dimensional) 
representation theory of this amalgam is (almost by construction)
$p$-locally determined. In the case of the fusion system associated 
to a $p$-block of a finite group, we suggest strong links between 
block-theoretic  invariants of the above finite epimorphic images 
of the associated  amalgam and of the original block. We believe that 
the results of this paper offer the prospect of at least the beginnings 
of a  structural explanation for some of the current numerical conjectures 
in block theory (eg [2],[8]).