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Some non-existence results on divisible difference sets
Authors:K. T. Arasu  James Davis  Dieter Jungnickel  Alexander Pott
Affiliation:1. Department of Mathematics and Statistics, Wright State University, 45435, Dayton, Ohio, USA
2. Dept. of Mathematics and Computer Science, University of Richmond, 23173, Richmond, Virginia, USA
3. Mathematisches Institut, Justus-Liebig-Universit?t Giessen, Arndtstr. 2, D-6300, Giessen, Germany
Abstract:In this paper, we shall prove several non-existence results for divisible difference sets, using three approaches:
  1. character sum arguments similar to the work of Turyn [25] for ordinary difference sets,
  2. involution arguments and
  3. 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.
Keywords:
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