The Mobius function of the small Ree groups
The M\"obius 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 M\"obius function of the simple small Ree groups, $R(q)^2G_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 \vert$Epi$\Gamma,G)\vert$ for various \Gamma$ such as $F_2$ or the modular group $PSL_2(\mathbb{Z} with applications to Grothendieck's theory of dessins d'enfants as well as probabilistic generation of the small Ree groups.
Australas. J Comb.