Projektivitaeten zwischen abelschen und nichtabelschen Gruppen

In 1951 S. Sato showed that the lattice of subgroups of a modular group with elements of infinite order is isomorphic to the one of a convenient abelian group. Recently in the last part of Sato's work some inexactitudes were found which could question the validity of the result. \par In this paper a new proof of that theorem is provided. Included are also two results on modular groups with elements of infinite order. Namely that any such group can be embedded in a modular group whose torsion- subgroup is divisible and that in case the torsion-subgroup is divisible a modular group splits into the semidirect product of its torsion- subgroup by a cyclic, or locally cyclic, group.