Abstract
LetA={a 1, …,a k} and {b 1, …,b k} be two subsets of an abelian groupG, k≤|G|. Snevily conjectured that, when |G| is odd, there is a numbering of the elements ofB such thata i+b i,1≤i≤k are pairwise distinct. By using a polynomial method, Alon affirmed this conjecture for |G| prime, even whenA is a sequence ofk<|G| elements. With a new application of the polynomial method, Dasgupta, Károlyi, Serra and Szegedy extended Alon’s result to the groupsZ rp andZ p rin the casek<p and verified Snevily’s conjecture for every cyclic group. In this paper, by employing group rings as a tool, we prove that Alon’s result is true for any finite abelianp-group withk<√2p, and verify Snevily’s conjecture for every abelian group of odd order in the casek<√p, wherep is the smallest prime divisor of |G|.
Similar content being viewed by others
References
N. Alon,Combinatorial Nullstellensatz, Probability and Computing8 (1999), 7–29.
N. Alon,Additive Latin transversals, Israel Journal of Mathematics117 (2000), 125–130.
S. Dasgupta, G. Károlyi, O. Serra and B. Szegedy,Transversals of additive Latin squares, Israel Journal of Mathematics126 (2001), 17–28.
W. D. Gao,Addition theorems and group rings, Journal of Combinatorial Theory. Series A77 (1997), 98–109.
C. Peng,Addition theorems in elementary abelian groups I, II, Journal of Number Theory27 (1987), 46–57, 58–62.
H. Snevily,The Cayley addition table of Z n,The American Mathematical Monthly106 (1999), 584–585.
Author information
Authors and Affiliations
Additional information
This work has been supported partly by NSFC grant number 19971058 and 10271080.
Rights and permissions
About this article
Cite this article
Gao, W.D., Wang, D.J. Additive Latin transversals and group rings. Isr. J. Math. 140, 375–380 (2004). https://doi.org/10.1007/BF02786641
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02786641