Abstract
In this paper we develop a new method to obtain identities in a group algebraGF(p)G if an abelian difference set of ordern≡0 (modp) exists inG. We give an explicit formula ifp 2 orp 3 is the exactp-power dividingn. This generalizes the approach of Wilbrink, Arasu and the author. The proof presented here uses some knowledge about field extensions of thep-adic numbers.
Similar content being viewed by others
References
K. T. Arasu: On Wilbrink's theorem.J. Comb. Th. (A) 44 (1987), 156–158
K. T. Arasu: Another variation of Wilbrink's theorem,Ars Combinatoria 25 (1988), 107–109.
K. T. Arasu andA. Pott: Relative difference sets with multiplier 2,Ars Combinatoria 27 (1989), 139–142.
T. Beth, D. Jungnickel andH. Lenz Design Theory, Bibliographisches Institut, Mannheim, 1985; Cambridge University Press, Cambridge, 1986.
B. Huppert:Endliche Gruppen I, Springer, Berlin, 1967.
D. Jungnickel: An elementary proof of Wilbrink's theorem,Archiv Math. 52 (1989), 615–617.
E. S. Lander:Symmetric Designs: An Algebraic Approach, Cambridge University Press, Cambridge, 1983.
A. Pott: Applications of the DFT to abelian difference sets,Archiv Math. 51 (1988), 283–288.
A. Pott: An affine analogue of Wilbrink's theorem,J. Comb. Th. (A) 55 (1990), 313–315.
H. A. Wilbrink: A note on planar difference sets,J. Comb. Th. (A) 38 (1985), 94–95.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pott, A. New necessary conditions on the existence of abelian difference sets. Combinatorica 12, 89–93 (1992). https://doi.org/10.1007/BF01191207
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01191207