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LetF  q\mathbb{F}_{q}    be the finite field with  elements and consider the -dimensionalF  q\mathbb{F}_{q}   -vector spaceV  =   F  q  nV=\mathbb{F}_{q}^{n}   . In this paper, we define a closure operator on the subgroup lattice of the groupG  =   PGL  ⁢   (  V  )G=\mathrm{PGL}(V)   . Let  denote the Möbius function of this lattice. The aim is to use this closure operator to characterize subgroups  of  for whichμ  ⁢   (  H  ,  G  )    ≠  0\mu(H,G)\neq 0   . Moreover, we establish a polynomial bound on the numberc  ⁢   (  m  )c(m)    of closed subgroups  of index  in  for which the lattice of -invariant subspaces of  is isomorphic to a product of chains. This bound depends only on  and not on the choice of  and . It is achieved by considering a similar closure operator for the subgroup lattice ofGL  ⁢   (  V  )\mathrm{GL}(V)    and the same results proven for this group.

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