  Three-dimensional $Z(N)$ lattice gauge theories at zero temperature are
studied for various values of $N$. Using a modified phenomenological
renormalization group, we explore the critical behavior of the generalized
$Z(N)$ model for $N=2,3,4,5,6,8$. Numerical computations are used to simulate
vector models for $N=2,3,4,5,6,8,13,20$ for lattices with linear extension up
to $L=96$. We locate the critical points of phase transitions and establish
their scaling with $N$. The values of the critical indices indicate that the
models with $N>4$ belong to the universality class of the three-dimensional
$XY$ model. However, the exponent $\alpha$ derived from the heat capacity is
consistent with the Ising universality class. We discuss a possible resolution
of this puzzle. We also demonstrate the existence of a rotationally symmetric
region within the ordered phase for all $N\geq 5$ at least in the finite
volume.
