M\"obius function of the subgroup lattice of a finite group and Euler Characteristic

TL;DR

This paper explores the M"obius function on subgroup lattices of finite groups, relating it to Euler characteristics of associated simplicial complexes, especially for subgroups of linear groups over finite fields.

Contribution

It establishes a connection between the M"obius function on subgroup lattices and the Euler characteristic of simplicial complexes derived from these lattices, focusing on irreducible subgroups of linear groups.

Findings

Relation between M"obius function and Euler characteristic of simplicial complexes

Explicit formulas for the M"obius function in the context of linear groups

Insights into the structure of subgroup lattices of finite groups

Abstract

The M\"obius 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 M\"obius 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 $G$.