The structure of the abelian groups containing McFarland difference sets

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