A closure operator on the subgroup lattice of $\mathrm{G}\mathrm{L}(n,q)$ and $\mathrm{P}\mathrm{G}\mathrm{L}(n,q)$ in relation to the zeros of the Möbius 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 $G=\mathrm{P}\mathrm{G}\mathrm{L}(V)$. Let $\mu$ denote the Möbius function of this lattice. The aim is to use this closure operator to characterize subgroups $H$ of $G$ for which $\mu (H,G)\mathrm{\ne }0$. 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 $\mathrm{G}\mathrm{L}(V)$ and the same results proven for this group.