Given a finite group G, let C(G) be the set of all cosets of all proper subgroups of G, ordered by inclusion. In joint work with Russ Woodroofe, we show that the order complex of C(G) is not acyclic in characteristic two, and therefore not contractible. This answers a question of K. S. Brown. Our proof uses P. A. Smith Theory and the Classification of Finite Simple Groups. From our proof, we are led to the following elementary problem on binomial coefficients, which remains open. Given a positive integer n, must there exist primes p,r such that every binomial coefficient n choose k (0 < k < n) is divisible by at least one of p,r? I will discuss our result and its proof, along with other problems in group theory that are closely related to the binomial coefficient problem.