The M\"obius function of ${\rm PSU}(3,2^{2^n})$

TL;DR

This paper computes the M"obius function for subgroups of the simple group ${ m PSU}(2,2^{2^n})$ and analyzes the structure of its subgroup lattice and conjugacy classes, providing new insights into its combinatorial properties.

Contribution

It introduces explicit calculations of the M"obius function for subgroups of ${ m PSU}(2,2^{2^n})$ and evaluates the Euler characteristic of the poset of p-subgroups.

Findings

Explicit M"obius function values for subgroup lattice

Euler characteristic of p-subgroup posets computed

Structural insights into conjugacy class posets

Abstract

Let $G$ be the simple group ${\rm PSU}(2,2^{2^n})$, $n>0$. For any subgroup $H$ of $G$, we compute the M\"obius function $\mu_L(H,G)$ of $H$ in the subgroup lattice $L$ of $G$, and the M\"obius function $\mu_{\bar L}([H],[G])$ of $[H]$ in the poset $\bar L$ of conjugacy classes of subgroups of $G$. For any prime $p$, we provide the Euler characteristic of the order complex of the poset of $p$-subgroups of $G$.