  We consider the quantum mechanics of an electron confined to move on an
infinite cylinder in the presence of a uniform radial magnetic field. This
problem is in certain ways very similar to the corresponding problem on the
infinite plane. Unlike the plane however, the group of symmetries of the
magnetic field, namely, rotations about the axis and the axial translations, is
{\em not} realized by the quantum electron but only a subgroup comprising
rotations and discrete translations along the axial direction, is. The basic
step size of discrete translations is such that the flux through the `unit
cylinder cell' is quantized in units of the flux quantum. The result is derived
in two different ways: using the condition of projective realization of
symmetry groups and using the more familiar approach of determining the
symmetries of a given Hamiltonian.
