The structure of the abelian groups containing McFarland difference sets

Abstract

A McFarland difference set is a difference set with parameters ( νv, k, λ) = ( q d + 1 ( q d + q d − 1 + ⋯ + q + 2), q d ( q d + q d − 1 + ⋯ + q + 1), q d ( q d − 1 + q d − 2 + ⋯ + q + 1)), where q = p f and p is a prime. Examples for such difference sets can be obtained in all groups of G which contain a subgroup E ≅ EA( q d + 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( q d + 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 {2 f − 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.