Subgroup families controlling $p$-local finite groups

by C. Broto, N. Castellana, J. Grodal, R. Levi, B. Oliver

A $p$-local finite group consists of a finite $p$-group $S$, together with 
a pair of categories which encode ``conjugacy'' relations among subgroups 
of $S$, and which are modelled on the fusion in a Sylow $p$-subgroup of a 
finite group.  It contains enough information to define a classifying 
space which has many of the same properties as $p$-completed  classifying 
spaces of finite groups.  In this paper, we examine which subgroups 
control this structure.  More precisely, we prove that the question of 
whether an abstract fusion system $F$ over a finite $p$-group $S$ is 
saturated can be determined by just looking at smaller classes of 
subgroups of $S$.  We also prove that the homotopy type of the classifying 
space of a given $p$-local finite group is independent of the family of 
subgroups used to define it, in the sense that it remains unchanged when 
that family ranges from the set of $F$-centric $F$-radical subgroups (at a 
minimum) to the set of $F$-quasicentric subgroups (at a maximum).  
Finally, we look at constrained fusion systems, analogous to 
$p$-constrained finite groups, and prove that they in fact all arise from 
groups.