Divisible Difference Sets With Multiplier -1
Document Type
Article
Publication Date
8-15-1990
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Abstract
We investigate proper (m, n, k, λ1, λ2)-divisible difference sets D in an abelian group G admitting the multiplier − 1. We show that this assumption implies severe restriction on the parameters of D and the structure of G. For instance, if D is even reversible (i.e., D is fixed by the multiplier − 1), the square-free part of k − λ1 has to be 1 or 2. In the case of relative difference sets (i.e., in the case λ1 = 0), one necessarily has k = m = nλ2, and thus the associated symmetric divisible design dev D has to be a symmetric transversal design. We also construct some new series of examples, among them an infinite series of relative difference sets D with “weak” multiplier − 1 (i.e., − 1 fixes no translate of D but still induces an automorphism of dev D—a situation which cannot arise for ordinary difference sets). Finally, we partially characterize the (reversible) divisible difference sets with k−λ1⩽1; moreover, we obtain a complete characterization of all cyclic reversible divisible difference sets for which n is even.
Repository Citation
Arasu, K. T.,
Jungnickel, D.,
& Pott, A.
(1990). Divisible Difference Sets With Multiplier -1. Journal of Algebra, 133 (1), 35-62.
https://corescholar.libraries.wright.edu/math/589
DOI
10.1016/0021-8693(90)90067-X
