Snevily's conjecture

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This article is about a conjecture in the following area in/related to group theory: additive combinatorics. View all conjectures and open problems

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

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.

References

Journal references

External links