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Home > Vol 27, No 2 (2008)

Omar A. AbuGhneim¹ and Ken W. Smith²

¹Jordan University Department of Mathematics, Faculty of Science Amman 11942 Jordan
²Central Michigan University Department of Mathematics Mount Pleasant MI 48859 USA

This work examines the existence of (4 q ²,2 q ² -  q, q ² -  q) difference sets, for q= p ^f , where p is a prime and f is a positive integer. Suppose that G is a group of order 4 q ² which has a normal subgroup K of order q such that G/ K \cong  C _q \times  C ₂\times  C ₂, where C _q , C ₂ are the cyclic groups of order q and 2 respectively. Under the assumption that p is greater than or equal to 5, this work shows that G does not admit (4 q ²,2 q ² -  q, q ² -  q) difference sets.

1. AbuGhneim, O.A., Becker, P.E., Mendes, J.K., Smith, K.W.: On Menon-Hadamard difference sets in groups of order 4p2. In: Proceedings of the 36th Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congresus Numerantium, vol. 172, pp. 97-121 (2005).
2. Beth, T., Jungnickel, D., Lenz, H.: Design Theory. Cambridge University Press, Cambridge (1986)
3. Curtis, C.W., Reiner, I.: Representation Theory of Finite Groups and Associative Algebras. Wiley Interscience, New York (1988)
4. Davis, J.A., Jedwab, J.: A survey of Hadamard difference sets. In: Arasu, K.T. (ed.) Groups, Differ- ence Sets and the Monster, pp. 145-156. de Gruyter, Berlin (1996)
5. Dillon, J.F.: Variatios on a scheme of McFarland for noncyclic difference sets. J. Comb. Theory Ser.