Snevily's conjecture: Difference between revisions

Revision as of 23:18, 1 December 2008

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 $φ : A \to B$ such that the sums $a + φ (a)$ 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

Snevily's conjecture has been proved for groups of 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}

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 ==Statement==
-Let <math>G</math> be an [[odd-order Abelian group]] and <math>A, B</math> be subsets of <math>G</math> of equal cardinality. Then, there is a bijection <math>\varphi:A \to B</math> such that the sums <math>a + \varphi(a)</math> are distinct for all <math>a \in A</math>.
+Let <math>G</math> be an [[fact about::odd-order Abelian group]] and <math>A, B</math> be subsets of <math>G</math> of equal cardinality. Then, there is a bijection <math>\varphi:A \to B</math> such that the sums <math>a + \varphi(a)</math> are distinct for all <math>a \in A</math>.
 ==Relation with other conjectures==