Möbius function of the subgroup lattice of a finite group and Euler characteristic

The Möbius function of the subgroup lattice of a finite group has been introduced by Hall and applied to investigate several questions. In this paper, we consider the Möbius function defined on an order ideal related to the lattice of the subgroups of an irreducible subgroup G of the general linear group \(\textrm{GL}(n,q)\) $\text{GL}(n,q)$ acting on the n-dimensional vector space \(V=\mathbb {F}_q^n\) $V={\mathbb{F}}_q^n$ , where \(\mathbb {F}_q\) ${\mathbb{F}}_q$ is the finite field with q elements. We find a relation between this function and the Euler characteristic of two simplicial complexes \(\Delta _1\) ${\mathrm{\Delta }}_1$ and \(\Delta _2\) ${\mathrm{\Delta }}_2$ , the former raising from the lattice of the subspaces of V, the latter from the subgroup lattice of G.