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

Abstract

Abstract Let F q \mathbb{F}_{q} be the finite field with 𝑞 elements and consider the 𝑛-dimensional F q \mathbb{F}_{q} -vector space V = F q n V=\mathbb{F}_{q}^{n} . In this paper, we define a closure operator on the subgroup lattice of the group G = PGL ⁢ ( V ) G=\mathrm{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 \mu(H,G)\neq 0 . Moreover, we establish a polynomial bound on the number c ⁢ ( m ) 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 ) \mathrm{GL}(V) and the same results proven for this group.