In this paper, we shall prove several non-existence results for divisible difference sets, using three approaches:
(i) character sum arguments similar to the work of Turyn  for ordinary difference sets,
(ii) involution arguments, and
(iii) multipliers in conjunction with results on ordinary difference sets.
Among other results, we show that an abelian affine difference set of odd order s (s not a perfect square) in G can exist only if the Sylow 2-subgroup of G is cyclic. We also obtain a non-existence result for non-cyclic (n, n, n, 1) relative difference sets of odd order n.
Copyright © 1991, Springer. This article first appeared in Combinatorica: 11:1 (1991), 1-8.
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Arasu, K. T., James A. Davis, Dieter Jungnickel, and Alexander Pott. "Some Non-Existence Results on Divisible Difference Sets." Combinatorica 11, no. 1 (March 1991): 1-8. doi:10.1007/BF01375467.