  We set an upper bound on the gravitational cutoff in theories with exact
quantum numbers of large N periodicity, such as Z_N discrete symmetries. The
bound stems from black hole physics. It is similar to the bound appearing in
theories with N particle species, though a priori, a large discrete symmetry
does not imply a large number of species. Thus, there emerges a potentially
wide class of new theories, that address the hierarchy problem by lowering the
gravitational cutoff due to existence of large Z_{10^32}-type symmetries.
