On the relationship between Monstrous Moonshine and the uniqueness of the Moonshine Module

We consider the relationship between the conjectured uniqueness of the Moonshine Module,, and Monstrous Moonshine, the genus zero property of the modular invariance group for each Monster group Thompson series. We first discuss a family of possibleZ _n meromorphic orbifold constructions of based on automorphisms of the Leech lattice compactified bosonic string. We reproduce the Thompson series for all 51 non-Fricke classes of the Monster groupM together with a new relationship between the centralisers of these classes and 51 corresponding Conway group centralisers (generalising a well-known relationship for 5 such classes). Assuming that is unique, we consider meromorphic orbifoldings of and show that Monstrous Moonshine holds if and onlyZ _r if the only meromorphic orbifoldings of are itself or the Leech theory. This constraint on the meromorphic orbifoldings of therefore relates Monstrous Moonshine to the uniqueness of in a new way.