# The M\"obius function of ${\rm PSL}(3,2^p)$ for any prime $p$

## TL;DR

This paper computes the M"obius function for subgroups of the simple group ${
m PSL}(3,2^p)$, providing detailed subgroup intersection descriptions and exploring connections with combinatorial objects and topological invariants.

## Contribution

It introduces explicit calculations of the M"obius function for all subgroups of ${
m PSL}(3,2^p)$ and describes subgroup intersections, extending understanding of subgroup lattice structures.

## Key findings

- Computed the M"obius function for all subgroups of ${
m PSL}(3,2^p)$
- Described intersections of maximal subgroups of ${
m PSL}(3,2^p)$
- Connected the M"obius function to combinatorial and topological invariants

## Abstract

Let $G$ be the simple group ${\rm PSL}(3,2^p)$, where $p$ is a prime number. For any subgroup $H$ of $G$, we compute the M\"obius function of $H$ in the subgroup lattice of $G$. To this aim, we describe the intersections of maximal subgroups of $G$. We point out some connections of the M\"obius function with other combinatorial objects, and, in this context, we compute the reduced Euler characteristic of the order complex of the subposet of $r$-subgroups of ${\rm PGL}(3,q)$, for any prime $r$ and any prime power $q$.