#### Abstract

Which groups *G* contain difference sets with the parameters (*v*, *k*, *λ*)= (*q*^{3} + 2*q*^{2} , *q*^{2} + *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 *q*^{2} 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 2*q* when *p* = 2 and at most *q* if *p* is an odd prime. In this paper we lower these exponent bounds when *q* ≠ *p* by showing that a difference set cannot exist for the bounding exponent values of 2*q* 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,* *λ*)-parameter values.

#### Document Type

Book Chapter

#### Publication Date

1996

#### Publisher Statement

Copyright © 1996 Walter de Gruyter. This chapter first appeared in* Groups, Difference Sets, and the Monster: Proceedings of a Special Research Quarter at the Ohio State University, Spring 1993*.

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#### Recommended Citation

Arasu, K. T., James A. Davis, Jonathan Jedwab, Siu Lun Ma, and Robert L. McFarland. "Exponent Bounds for a Family of Abelian Difference Sets." In *Groups, Difference Sets, and the Monster: Proceedings of a Special Research Quarter at the Ohio State University, Spring 1993*, edited by K. T. Arasu, J. F. Dillon, K. Harada, S. Sehgal, and R. Solomon, 129-43. Ohio State University Mathematical Research Institute Publications. New York: Walter De Gruyter, 1996.