# 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 $a \in A$.

## Progress towards the conjecture

### For subsets of size two

Further information: Snevily's 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

Further information: Snevily's conjecture 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.