The propagation of a nonrelativistic electron beam injected into space plasma from a conductor was studied by means of an electrostatic three-dimensional simulation model. Only the injection of an electron beam along a magnetic field was considered. Contrary to one- and two-dimensional simulations, an overdense electron beam whose density is much larger than the ambient plasma density can escape a spacecraft without causing significant charging. It was found that the radial electric field arising from the space charge of the beam electrons can give rise to a rotation of both the beams and ambient electrons with respect to ambient ions. The rotation speed can easily exceed the electron thermal speed even when the beam density is smaller than the ambient density. Instabilities arising from such rotating electron beam are discussed.