The Möbius function of the small Ree groups

Abstract

The Möbius function for a group, $G$, was introduced in 1936 by Hall in order to count ordered generating sets of $G$. In this paper we determine the Möbius function of the simple small Ree groups, $R(q)={}^2{G}_2(q)$ where $q=3^{2m+1}$ for $m>0$, using their 2-transitive permutation representation of degree $q^3+1$ and describe their maximal subgroups in terms of this representation. We then use this to determine $|$Epi$(\Gamma ,G)|$ for various $\Gamma$, such as $F_2$ or the modular group $PS{L}_2(\mathbb{Z})$, with applications to Grothendieck's theory of dessins d'enfants as well as probabilistic generation of the small Ree groups.

Includes the determination of the Möbius function for various finitely presented groups, such as $F_2$ and $PS{L}_2(\mathbb{Z})$ with applications to probabilistic generation