The Mobius function of the small Ree groups

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) = G2(q) where q = 3 for m > 0, using their 2-transitive permutation representation of degree q + 1. We also describe their maximal subgroups in terms of this representation. We use this to enumerate smooth epimorphisms from Γ to G for various finitely presented groups Γ, such as F2 and the modular group PSL2(Z). We then highlight applications of these enumerations to Grothendieck’s theory of dessins d’enfants as well as probabilistic generation of the small Ree groups.