The Mobius function of the small Ree groups

The Mobius 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 Mobius function of the simple small Ree groups, R(q) = G2(q) where q = 32m+1 for m > 0, using their 2-transitive permutation representation of degree q3 + 1 and describe their maximal subgroups in terms of this representation. We then use this to determine |Epi(Γ, G)| for various Γ, such as F2 or the modular group PSL2(Z), with applications to Grothendieck’s theory of dessins d’enfants as well as probabilistic generation of the small Ree groups.