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

TL;DR

This paper introduces a closure operator on the subgroup lattice of projective general linear groups over finite fields, linking it to the M"obius function zeros and providing bounds on certain subgroup counts.

Contribution

It defines a new closure operator on subgroup lattices of $ ext{PGL}(n,q)$ and $ ext{GL}(n,q)$, characterizing subgroups with non-zero M"obius function and bounding the number of specific closed subgroups.

Findings

Characterization of subgroups with non-zero M"obius function values.

Polynomial bounds on the number of closed subgroups of a given index.

Extension of results from $ ext{GL}(V)$ to $ ext{PGL}(V)$.

Abstract

Let $\mathbb{F}_q$ be the finite field with $q$ elements and consider the $n$-dimensional $\mathbb{F}_q$-vector space $V=\mathbb{F}_q^n\,$. In this paper we define a closure operator on the subgroup lattice of the group $G = \mathrm{PGL}(V)$. Let $\mu$ denote the M\"obius function of this lattice. The aim is to use this closure operator to characterize subgroups $H$ of $G$ for which $\mu(H,G)\neq 0$. Moreover, we establish a polynomial bound on the number $c(m)$ 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 $\mathrm{GL}(V)$ and the same results proven for this group.