The Möbius function of $\mathrm{PSU}(3,2^{2^n})$

Abstract

Let $G$ be the simple group $\mathrm{PSU}(2,2^{2^n})$, $n>0$. For any subgroup $H$ of $G$, we compute the Möbius function ${\mu }_L(H,G)$ of $H$ in the subgroup lattice $L$ of $G$, and the Möbius 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 $G$.