We study the binary star formation process from a rotating magnetized molecular cloud. We assume an isothermal cylindrical cloud in hydrostatic balance whose rotation axis and the direction of global magnetic field lines are both identical, and parallel to the cylinder axis. We added axisymmetric and non-axisymmetric density perturbations to the initial state and followed the subsequent evolutions. The evolution is characterized by three parameters: the amplitude of the non-axisymmetric perturbations, the rotation speed, and the magnetic field strength. As a result, it is found that non-axisymmetry hardly evolves in the early phase, but begins to grow after the gas contracts and forms a thin disk. There are two types of fragmentation: fragmentation from a ring and that from a bar. Thin adiabatic cores fragments if a thickness is smaller than 1/4 of the radius. For the fragments to survive, they should be formed in a heavily elongated barred core or a flat round disk. In the models showing fragmentation, outflows from respective fragments are found as well as those driven by the rotating bar or the disk.