  The instability of a supercritical Taylor-Couette flow of a conducting fluid
with resting outer cylinder under the influence of a uniform axial electric
current is investigated for magnetic Prandtl number Pm=1. In the linear theory
the critical Reynolds number for axisymmetric perturbations is not influenced
by the current-induced axisymmetric magnetic field but all axisymmetric
magnetic perturbations decay. The nonaxisymmetric perturbations with m=1 are
excited even without rotation for large enough Hartmann numbers ("Tayler
instability"). For slow rotation their growth rates scale with the Alfv\'en
frequency of the magnetic field but for fast rotation they scale with the
rotation rate of the inner cylinder. In the nonlinear regime the ratio of the
energy of the magnetic m=1 modes and the toroidal background field is very low
for the non-rotating Tayler instability but it strongly grows if differential
rotation is present. For super-Alfv\'enic rotation the energies in the m=1
modes of flow and field do not depend on the molecular viscosity, they are
almost in equipartition and contain only 1.5% of the centrifugal energy of the
inner cylinder. The geometry of the excited magnetic field pattern is strictly
nonaxisymmetric for slow rotation but it is of the mixed-mode type for fast
rotation -- contrary to the situation which has been observed at the surface of
Ap stars.
