Let G be a finite group and let C(G) be the set of all cosets of all proper subgroups of G, partially ordered by inclusion. The order complex of C(G) is the abstract simplicial complex whose faces are the totally ordered subsets of C(G). K. S. Brown asked whether this complex can be contractible. In joint work with Russ Woodroofe, we show that this complex cannot be acyclic unless some composition factor of G is a very large alternating group. We expect that in fact the complex is never acyclic. The attempt to extend our method to arbitrary finite groups leads to an elementary but seemingly difficult number theoretic problem.