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Title: | The structure of the abelian groups containing McFarland difference sets | Authors: | Ma, S.L. Schmidt, B. |
Issue Date: | May-1995 | Citation: | Ma, S.L.,Schmidt, B. (1995-05). The structure of the abelian groups containing McFarland difference sets. Journal of Combinatorial Theory, Series A 70 (2) : 313-322. ScholarBank@NUS Repository. | Abstract: | A McFarland difference set is a difference set with parameters (νv, k, λ) = (qd + 1(qd + qd - 1 + ⋯ + q + 2), qd(qd + qd - 1 + ⋯ + q + 1), qd(qd - 1 + qd - 2 + ⋯ + q + 1)), where q = pf and p is a prime. Examples for such difference sets can be obtained in all groups of G which contain a subgroup E ≅ EA(qd + 1) such that the hyperplanes of E are normal subgroups of G. In this paper we study the structure of the Sylow p-subgroup P of an abelian group G admitting a McFarland difference set. We prove that if P is odd and P is self-conjugate modulo exp(G), then P ≅ EA(qd + 1). For p = 2, we have some strong restrictions on the exponent and the rank of P. In particular, we show that if f ≥ 2 and 2 is self-conjugate modulo exp(G), then exp(P) ≤ max {2f - 1, 4}. The possibility of applying our method to other difference sets has also been investigated. For example, a similar method is used to study abelian (320, 88, 24)-difference sets. © 1995. | Source Title: | Journal of Combinatorial Theory, Series A | URI: | http://scholarbank.nus.edu.sg/handle/10635/104364 | ISSN: | 00973165 |
Appears in Collections: | Staff Publications |
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