  We investigate the phase diagram and the nature of the phase transitions of
three-dimensional lattice gauge-Higgs models obtained by gauging the Z_N
subgroup of the global Z_q invariance group of the Z_q clock model (N is a
submultiple of q). The phase diagram is generally characterized by the presence
of three different phases, separated by three distinct transition lines. We
investigate the critical behavior along the two transition lines characterized
by the ordering of the scalar field. Along the transition line separating the
disordered-confined phase from the ordered-deconfined phase, standard arguments
within the Landau-Ginzburg-Wilson framework predict that the behavior is the
same as in a generic ferromagnetic model with Z_p global symmetry, p being the
ratio q/N. Thus, continuous transitions belong to the Ising and to the O(2)
universality class for p=2 and p>3, respectively, while for p=3 only
first-order transitions are possible. The results of Monte Carlo simulations
confirm these predictions. There is also a second transition line, which
separates two phases in which gauge fields are essentially ordered. Along this
line we observe the same critical behavior as in the Z_q clock model, as it
occurs in the absence of gauge fields.
