For a prime p, a homology decomposition of the classifying space BG of a finite group G
consist of a functor F : D → spaces from a small category into the category of spaces
and a map hocolim F → BG from the homotopy colimit to BG that induces an isomorphism
in mod-p homology. Associated to a modular representation
G → Gl(n; [ ]p), a family of subgroups is constructed that is
closed under conjugation, which gives rise to three different homology decompositions, the so-called
subgroup, centralizer and normalizer decompositions. For an action of G on an [ ]p-vector
space V, this collection consists of all subgroups of G with nontrivial p-Sylow subgroup
which fix nontrivial (proper) subspaces of V pointwise. These decomposition formulas connect the modular
representation theory of G with the homotopy theory of BG.