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.