The normaliser decomposition for $p$–local finite groups

Abstract

We construct an analogue of the normaliser decomposition for $p$–local finite groups $\left(S,\mathcal{ℱ},\mathcal{ℒ}\right)$ with respect to collections of $\mathcal{ℱ}$–centric subgroups and collections of elementary abelian subgroups of $S$. This enables us to describe the classifying space of a $p$–local finite group, before $p$–completion, as the homotopy colimit of a diagram of classifying spaces of finite groups whose shape is a poset and all maps are induced by group monomorphisms.