Homotopy equivalence of posets with a group action

The purpose of the present paper is to provide proofs of some of the results announced in the survey paper [13]. These results deal with combinatorial topology, in particular with complexes associated with posets of subgroups of a finite group. For applications to group theory, the reader can refer to the above mentioned paper [13]. In his influential paper [11], Quillen proved that the poset Sp(G) consisting of the non-identity p-subgroups of a finite group G is homotopy equivalent to its subposet Ap(G) consisting of the non-identity elementary abelian p-subgroups. Subsequently Bouc [2] proved in a dual fashion that Sp(G) is also homotopy equivalent to the subposet