Abstract
In this paper, we shall prove several non-existence results for divisible difference sets, using three approaches:
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(i)
character sum arguments similar to the work of Turyn [25] for ordinary difference sets,
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(ii)
involution arguments and
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(iii)
multipliers in conjunction with results on ordinary difference sets.
Among other results, we show that an abelian affine difference set of odd orders (s not a perfect square) inG can exist only if the Sylow 2-subgroup ofG is cyclic. We also obtain a non-existence result for non-cyclic (n, n, n, 1) relative difference sets of odd ordern.
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References
K. T. Arasu: “On Wilbrink's theorem”,J. Comb. Th. Ser. A44, (1987) 156–158.
K. T. Arasu: “Another variation of Wilbrink's theorem”,Ars Combinatoria,25 (1988), 107–109.
K. T. Arasu: “Cyclic affine planes of even order”,Disc. Math.,76 (1989), 177–181.
K. T. Arasu, andD. Jungnickel: “Affine difference sets of even order”,J. Comb. Th. A. 52 (1989), 188–196.
K. T. Arasu, andA. Pott: “Relative difference sets with multipliers 2”,Ars Combinatoria,27, (1989), 139–142.
T. Beth, D. Jungnickel, andH. Lenz:Design Theory, Bibliographisches Institut, Mannheim (1985), and Cambridge University Press, Cambridge (1986).
R. C. Bose: “An affine analogue of Singer's theorem”,J. Ind. Math. Soc. 6 (1942), 1–15.
W. de Launey: “On the non-existence of generalized weighing matrices”,Ars Combinatoria 17A (1984), 117–132.
J. E. H. Elliott, andA. T. Butso: “Relative difference sets”,Illinois J. Math. 10 (1966), 517–531.
M. J. Ganely: “On a paper of Dembowski and Ostrom”,Arch. Math. 27 (1976), 93–98.
D. Ghinelli-Smit: “A new result on difference sets with −1 as a multiplier”,Geom. Ded. 23 (1987), 309–317.
A. J. Hoffman: “Cyclic affine planes”,Can. J. Math. 4 (1952), 295–301.
D. R. Hughes: “Partial difference sets”,Amer. J. Math. 78 (1956), 650–674.
B. Huppert:Endliche Gruppen I, Springer-Verlag, Berlin-Heidelberg-New York, 1967.
E. C. Johnsen: “The inverse multiplier for abelian group difference sets”,Can. J. Math. 16 (1964), 787–796.
D. Jungnickel: “Difference sets with multiplier −1”,Arch. Math. 38 (1982), 511–512.
D. Jungnmickel: “On automorphism groups of divisible designs”,Can. J. Math. 34 (1982), 257–297.
D. Jungnickel: “A note on affine difference sets”,Arch. Math. 47 (1986), 279–280.
D. Jungnickel: “On a theorem of Ganley”,Graphs and Comb. 3 (1987), 141–143.
D. Jungnickel: “On automorphism groups of divisible designs, II: Group invariant generalized conference matrices”,Arch. Math. 54 (1990), 200–208.
H. P. Ko, andD. K. Ray-Chaudhuri: “Multiplier theorems”,J. Comb. Th. A 30 (1981), 134–157.
C. W. H. Lam: “On relative difference sets”, Proc. 7th Manitoba conference on numerical math. and computing, (1977), 445–474.
A. Pott: “Affine analogue of Wilbrink's theorem”,J. Comb. Th. A.55 (1990), 313–315.
A. Pott: “On abelian difference sets with multiplier −1,Arch. Math. 53 (1989), 510–512.
R. J. Turyn: “Character sums and difference sets”,Pacific J. Math. 15 (1965), 319–346.
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The first author's research was partially supported by NSA Grant #MDA 904-87-H-2018. The second and fourth authors gratefully acknowledge the hospitality of Wright State University during the time of this research. The last two authors thank the University of Waterloo for its hospitality, and the third author also acknowledges the financial support of NSERC under Grant #IS-0367.
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Arasu, K.T., Davis, J., Jungnickel, D. et al. Some non-existence results on divisible difference sets. Combinatorica 11, 1–8 (1991). https://doi.org/10.1007/BF01375467
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DOI: https://doi.org/10.1007/BF01375467