| In this thesis we will investigate combinatorial properties of partially ordered sets associated with finite groups. In 1989, Hawkes, Isaacs, and Ozaydin conjectured that the Mobius number of the subgroup lattice of a finite group G was equal to the Mobius number of the frame of G times the order of the derived subgroup G' of G. This conjecture is false in general. In this thesis we will verify that the conjecture holds for two dimensional projective general linear or projective special linear groups over a field of order p2n , where p is an odd prime and n is a natural number, and also for symmetric groups of degree p. We will do this by determining the Mobius number of the frames of the aforementioned groups and comparing this value with known values of the Mobius number of the subgroup lattices of these groups. We will also determine a formula for the Mobius number of the frame of the symmetric group of degree n that holds for all natural numbers n in terms of certain transitive subgroups of this group, as well as a formula for the Mobius number of the frame of the symmetric group of degree 2 p for p an odd prime in terms of certain primitive subgroups of this group. |
