Divisible difference sets with multiplier −1

Abstract

We investigate proper ( m, n, k, λ 1, λ 2)-divisible difference sets D in an abelian group G admitting the multiplier − 1. We show that this assumption implies severe restriction on the parameters of D and the structure of G. For instance, if D is even reversible (i.e., D is fixed by the multiplier − 1), the square-free part of k − λ 1 has to be 1 or 2. In the case of relative difference sets (i.e., in the case λ 1 = 0), one necessarily has k = m = nλ 2, and thus the associated symmetric divisible design dev D has to be a symmetric transversal design. We also construct some new series of examples, among them an infinite series of relative difference sets D with “weak” multiplier − 1 (i.e., − 1 fixes no translate of D but still induces an automorphism of dev D—a situation which cannot arise for ordinary difference sets). Finally, we partially characterize the (reversible) divisible difference sets with k− λ 1⩽1; moreover, we obtain a complete characterization of all cyclic reversible divisible difference sets for which n is even.