A closure operator on the subgroup lattice of GL(n, q) and PGL(n, q) in relation to the zeros of the Möbius function

Let Fq be the finite field with q elements and consider the n-dimensional Fq-vector space V = Fq^n. In this paper, we define a closure operator on the subgroup lattice of the group G = PGL(V). Let μ denote the Möbius function of this lattice. The aim is to use this closure operator to characterize subgroups H of G for which μ (H; G) ≠ 0. Moreover, we establish a polynomial bound on the number c(m) of closed subgroups H of index m in G for which the lattice of H-invariant subspaces of V is isomorphic to a product of chains. This bound depends only on m and not on the choice of n and q. It is achieved by considering a similar closure operator for the subgroup lattice of GL(V) and the same results proven for this group.