Transversals of additive latin squares

Let A = {a(1),...,a(k)} and B = {b(1),...,b(k)} be two subsets of an Abelian group G, k less than or equal to \G\. Snevily conjectured that, when G is of odd order, there is a permutation pi epsilon S-k such that the sums a(i) + b(pi(i)), 1 less than or equal to i less than or equal to k, are pairwise different. Alon showed that the conjecture is true for groups of prime order, even when A is a sequence of k < \G\ elements, i.e., by allowing repeated elements in A. In this last sense the result does not hold for other Abelian groups. With a new kind of application of the polynomial method in various finite and infinite fields we extend Alon's result to the groups (Z(p))(alpha) and Z(p)alpha in the case k < p, and verify Snevily's conjecture for every cyclic group of odd order.