  This is the first paper about the fragmentation and mass outflow in the
molecular cloud by using three-dimensional MHD nested-grid simulations. The
binary star formation process is studied paying particular attention to the
fragmentation of a rotating magnetized molecular cloud. We assume an isothermal
rotating and magnetized cylindrical cloud in hydrostatic balance.
Non-axisymmetric as well as axisymmetric perturbations are added to the initial
state and the subsequent evolutions are studied. 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. Disk formation is strongly promoted by the
rotation speed and the magnetic field strength. There are two types of
fragmentation: fragmentation from a ring and that from a bar. Thin adiabatic
cores fragment 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.
