Homotopy equivalences between p-subgroup categories

Let SG⁎ be the Brown poset of nonidentity p-subgroups of the finite group G   ordered by inclusion. Results of Bouc and Quillen show that SG⁎ is homotopy equivalent to its subposets SG⁎+rad of nonidentity radical p  -subgroups and SG⁎+eab of nonidentity elementary abelian p-subgroups. In this note we extend these results for the Brown poset of G to other categories of p-subgroups of G, including the p-fusion system of G.