In 1936 Hall showed that Möbius inversion could be applied to the lattice of subgroups of a finite group G in order to determine the number of n-bases of G, that is, generating sets of G of size n. The question can be modified and n-bases subject to certain relations can also be enumerated with applications to the theory of Riemann surfaces, Hurwitz groups, dessins d'enfants and various other algebraic, topological and combinatorial enumerations. In order to determine the Möbius function of a group it is necessary to understand the subgroup structure of a group and so we also give a description of the simple small Ree groups R(q)=2G2(q), in particular their maximal subgroups, in terms of their 2-transitive permutation representations of degree q3+1.
The Möbius function of the small Ree groups
Emilio Pierro (Birkbeck)
Mon, 10/11/2014 - 16:30