  The nonaxisymmetric Tayler instability of toroidal magnetic fields due to
axial electric currents is studied for conducting incompressible fluids between
two coaxial cylinders without endplates. The inner cylinder is considered as so
thin that even the limit of R_in \to 0 can be computed. The magnetic Prandtl
number is varied over many orders of magnitudes but the azimuthal mode number
of the perturbations is fixed to m=1. In the linear approximation the critical
magnetic field amplitudes and the growth rates of the instability are
determined for both resting and rotating cylinders. Without rotation the
critical Hartmann numbers do {\em not} depend on the magnetic Prandtl number
but this is not true for the growth rates. For given product of viscosity and
magnetic diffusivity the growth rates for small and large magnetic Prandtl
number are much smaller than those for Pm=1. For gallium under the influence of
a magnetic field at the outer cylinder of 1 kG the resulting growth time is 5
s. The minimum electric current through a container of 10 cm diameter to excite
the kink-type instability is 3.20 kA. For a rotating container both the
critical magnetic field and the related growth times are larger than for the
resting column.
