Snevily's conjecture

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

• Hall-Paige conjecture

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

• Additive Latin transversals by Noga Alon, , Volume 117, Page 125 - 130(Year 2000): ^{PDF copy (Springerlink)More info}
• Tranversals of additive Latin squares by Samit Dasgupta, Gyula Károlyi and Oriol Serra, , Volume 126, Page 17 - 28(Year 2001): ^{PDF copy (Springerlink)More info}

External links

• Open Problem Garden page