Daniel A. Ramras — Page 2

arXiv2002

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 further Brown's study of the homotopy type of the coset poset, 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 the fundamental group of L(G) in nearly all cases.

math.AT↗

arXiv2002

Connectivity of the coset poset

Topological concepts may be applied to any poset via the simplicial complex of finite chains. The coset poset C(G) of a finite group G (consisting of all cosets of all proper subgroups of G, ordered by inclusion) was introduced by Kenneth S. Brown in his study of the probabilistic zeta function P(G,s), and Brown's results motivate the study of the homotopy type of C(G). After a detailed introduction to basic simplicial topology and homotopy theory for finite posets (Chapter 2), we examine the connectivity of the coset poset (Chapter 3). Of particular interest is the fundamental group, and we provide various conditions under which C(G) is or is not simply connected. In particular, we classify direct products with simply connected coset posets. In Chapter 4, we treat several specific examples, including A_5 and PSL(2,7). We prove that both of these groups have simply connected coset posets, and in fact determine the precise homotopy type of C(A_5). This chapter also includes a detailed determination of all subgroups of the groups PSL(2,p), p prime (following Burnside). The final chapter discusses conjectures and directions for further research.

math.GR↗