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Reversible \((m,n,k,\lambda_ 1,\lambda_ 2)\)-abelian divisible difference sets with \(k-\lambda_ 1\) nonsquare. (English) Zbl 0833.05010

Let \(G\) be an abelian group of order \(mn\) with a subgroup \(N\) of order \(n\). A \(k\)-subset \(D\) of \(G\) such that \(D= D^{(- 1)}\) is a reversible \((m, n, k, \lambda_1, \lambda_2)\)-DDS in \(G\) relative to \(N\) iff \(D^2= k- \lambda_1+ \lambda_1 N+ \lambda_2(G\smallsetminus N)\). Suppose \(k- \lambda_1\) is nonsquare, \(D\) is nondegenerate, and \(p\) is an odd prime. If \(p|k- \lambda_1\) or \(p|k^2- \lambda_2mn\) then \(p|m\); if \(p|m\) then \(p|k^2- \lambda_2 mn\). Moreover, if \(D\) is proper then \((m, n, k, \lambda_1, \lambda_2)\) can writen as \((4m_1, 2^a, 2^a k_1, 2^a\mu_1, 2^{a- 1} \mu_2)\).
Reviewer: G.Ferrero (Parma)

MSC:

05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
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[1] Arasu, K. T.; Davis, J. A.; Jungnickel, D.; Pott, A., Some non-existence results on divisible difference sets, Combinatorica, 11, 1, 1-8 (1991) · Zbl 0765.05018
[2] Arasu, K. T.; Jungnickel, D.; Pott, A., Divisible difference sets with multiplier - 1, J. Algebra, 133, 35-62 (1990) · Zbl 0706.05012
[3] Arasu, K. T.; Ray-Chaudhuri, D. K., Multiplier theorem for difference lists, Ars Combin., 22, 119-137 (1986) · Zbl 0615.05016
[4] Beth, T.; Jungnickel, D.; Lenz, H., Design Theory (1985), Bibliographsches Institut: Bibliographsches Institut Mannheim
[5] Johnsen, E. C., The inverse multiplier for abelian group difference sets, Canad. J. Math., 16, 787-798 (1964) · Zbl 0123.25105
[6] Ko, H. P.; Ray-Chaudhuri, D. K., Multiplier theorems, J. Combin. Theory Ser. A, 30, 134-157 (1981) · Zbl 0479.05014
[7] Ko, H. P.; Ray-Chaudhuri, D. K., Intersection theorems for group divisible difference sets, Discrete Math., 39 (1982) · Zbl 0484.05021
[8] Lander, E. S., Symmetric design, an algebraic approach, London Math. Soc. Lecture notes ser., 74 (1983) · Zbl 0502.05010
[9] Lang, S., Algebra (1984), Addison-Wesley: Addison-Wesley Reading, MA
[10] Ka Hin Leung; Ma, S. L.; Tan, V., Abelian divisible difference sets with multiplier, I.J. Combin. Theory. Ser. A, 59, 51-72 (1992) · Zbl 0755.05010
[11] Ma, S. L., Partial difference sets, Discrete Math., 52, 75-89 (1984) · Zbl 0548.05012
[12] S.L. Ma, Polynomial addition sets, Ph.D. Thesis, University of Hong Kong.; S.L. Ma, Polynomial addition sets, Ph.D. Thesis, University of Hong Kong. · Zbl 0575.05036
[13] Ma, S. L., On divisible difference sets which are fixed by the inverse, Arch. Math., 54, 409-416 (1990) · Zbl 0685.05008
[14] McFarland, R. L.; Ma, S. L., Abelian difference sets with multiplier - 1, Arch. Math., 54, 610-623 (1990) · Zbl 0682.05018
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