Transversals of additive Latin squares

Let A = {a1,..., ak} and B = {b1,..., bk} be two subsets of an Abelian group G, k ≤ |G|. Snevily conjectured that, when G is of odd order, there is a permutation π ≤ Sk such that the sums ai + bπ(i), 1 ≤ i ≤ 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 (Zp)^α and Zpα in the case k < p, and verify Snevily's conjecture for every cyclic group of odd order.

Duke Scholars

Author Samit Dasgupta Mathematics