A Note on Difference Sets

LetDbe a (v, k, λ)-difference set in a groupG. Assume thatGhas a normal subgroupNsuch thatG/Nis cyclic or nearly cyclic. Under the self-conjugacy assumption on exp(G/N), we shall give bounds on |N| andλ. The theorem is applicable to a wider variety of parameters for groups, not necessarily abelian. These bounds exclude a (96, 20, 4)-difference set inZ/4Z×Z/8Z×Z/3ZorZ/2Z×Z/2Z×Z/8Z×Z/3Z, which were recently proved by Arasuet al. [1996, K. T. Arasu, J. A. Davis, J. Jedwab, S. L. Ma, and R. L. McFarland,in“Exponent Bounds for a Family of Abelian Difference Sets” (K. T. Arasu, J. F. Dillon, K. Harada, S. K. Sehgal, and R. L. Solomon, Eds.), pp. 129–143. DeGruyter Verlag, Berlin/New York].