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

Abstract

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 $\mathrm{GL}(n,q)$ acting on the $n$-dimensional vector space $V={\mathbb{F}}_q^n$, where ${\mathbb{F}}_q$ is the finite field with $q$ elements. We find a relation between this function and the Euler characteristic of two simplicial complexes ${\Delta }_1$ and ${\Delta }_2$, the former raising from the lattice of the subspaces of $V$, the latter from the subgroup lattice of $G$.

10 pages. Revised version; the first version contained some inaccuracies that have been corrected

References in corpus (1)

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