  The large hierarchy between the neutrino mass scale and that of the other
fermions seems to be unnatural from a theoretical point of view. Various
strategies have been devised in order to generate naturally small values of
neutrino masses. One of these techniques is neutrino mass generation at the
loop level which requires a mechanism, e.g., a symmetry, to forbid the lower
order contributions. Here, we study in detail the conditions on this type of
symmetries. We put special emphasis on the discrete Z_n symmetries as a simple
example but our results can be also extended to more general groups. We find
that regardless of the details of the symmetry, in certain cases the existence
of a lower order contribution to neutrino masses can be determined by the
topology of the diagrams with a given number of loops. We discuss the lepton
flavor violating rare decays as well as (g-2)_\mu in this class of models,
which generically appear at the one loop level. Typically the imposed symmetry
has important implications for dark matter, with the possibility of stabilizing
one or even multiple dark matter candidates.
