A closure operator on the subgroup lattice of $\mathrm{GL}(n,q)$ and $\mathrm{PGL}(n,q)$ in relation to the zeros of the M\'obius function

Let $\mathbb{F}_q$ be the finite field with $q$ elements and consider the $n$-dimensional $\mathbb{F}_q$-vector space $V=\mathbb{F}_q^n\,$. In this paper we define a closure operator on the subgroup lattice of the group $G = \mathrm{PGL}(V)$. Let $\mu$ denote the M\"obius function of this lattice. The aim is to use this closure operator to characterize subgroups $H$ of $G$ for which $\mu(H,G)\neq 0$. 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 $\mathrm{GL}(V)$ and the same results proven for this group.
Comment: 21 pages. Revised version; the exposition of the material has been improved compared to the first version (thanks to the suggestions of a reviewer)