Exponent Bounds for a Family of Abelian Difference Sets

Which groups G contain difference sets with the parameters (v, k, A.)= (q 3 + 2q 2 , q2 + q, q), where q is a power of a prime p? Constructions of K. Takeuchi, R.L. McFarland, and J.F. Dillon together yield difference sets with these parameters if G contains an elementary abelian group of order q2 in its center. A result of R.J. Turyn implies that if G is abelian and p is self-conjugate modulo the exponent of G, then a necessary condition for existence is that the exponent of the Sylow p-subgroup of G be at most 2q when p = 2 and at most q if p is an odd prime. In this paper we lower these exponent bounds when q =f. p by showing that a difference set cannot exist for the bounding exponent values of 2q and q. Thus if there exists an abelian (96, 20, 4)-difference set, then the exponent of the Sylow 2-subgroup is at most 4. We also obtain some nonexistence results for a more general family of (v, k, A.)-parameter values.