Snevily's conjecture

Statement
Let $$G$$ be an odd-order abelian group and $$A, B$$ be subsets of $$G$$ of equal cardinality. Then, there is a bijection $$\varphi:A \to B$$ such that the sums $$a + \varphi(a)$$ are distinct for all $$a \in A$$.

Relation with other conjectures

 * Hall-Paige conjecture

For subsets of size two
If $$A, B$$ are subsets of size two in an abelian group of odd order, Snevily's conjecture holds. This is easy to verify.

For cyclic groups
Snevily's conjecture has been proved for groups of odd prime order by Alon, and for all odd-order cyclic groups by Dasgupta, Karolyi, Serra, and Szegedy.