Transversals of additive Latin squares

Abstract

LetA={a 1, …,a k} andB={b 1, …,b k} be two subsets of an Abelian groupG, k≤|G|. Snevily conjectured that, whenG is of odd order, there is a permutationπ ∈S ksuch that the sums α i +b i , 1≤i≤k, are pairwise different. Alon showed that the conjecture is true for groups of prime order, even whenA is a sequence ofk<|G| elements, i.e., by allowing repeated elements inA. 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 ) a and\(\mathbb{Z}_{p^a } \) in the casek<p, and verify Snevily’s conjecture for every cyclic group of odd order.