  A subgroup H of G=(Z/dZ)^* is called balanced if every coset of H is evenly
distributed between the lower and upper halves of G, i.e., has equal numbers of
elements with representatives in (0,d/2) and (d/2,d). This notion has
applications to ranks of elliptic curves. We give a simple criterion in terms
of characters for a subgroup H to be balanced, and for a fixed integer p, we
study the distribution of integers d such that the cyclic subgroup of (Z/dZ)^*
generated by p is balanced.
