Connectivity of the coset poset and the subgroup poset of a group

We study the connectivity of the coset poset and the subgroup poset of a group, focusing in particular on simple connectivity. The coset poset was recently introduced by K. S. Brown in connection with the probabilistic zeta function of a group. We take Brown's study of the homotopy type of the coset poset further, and in particular generalize his results on direct products and classify direct products with simply connected coset posets. The homotopy type of the subgroup poset L(G) has been examined previously by Kratzer, Thevenaz, and Shareshian. We generalize some results of Kratzer and Thevenaz, and determine pi(1)(L(G)) in nearly all cases.