The Möbius function of $\mathrm{PSL}(3,2^p)$ for any prime $p$

Abstract

Let $G$ be the simple group $\mathrm{PSL}(3,2^p)$, where $p$ is a prime number. For any subgroup $H$ of $G$, we compute the Möbius 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 Möbius 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 $\mathrm{PGL}(3,q)$, for any prime $r$ and any prime power $q$.