ABELIAN DIVISIBLE DIFFERENCE SETS WITH MULTIPLIER -1

We investigate (m, n, k, λ1, λ2)-divisible difference sets in an abelian group admitting -1 as a multiplier. For the reversible case, we show that this assumption implies severe restrictions on the structure of divisible difference sets. In particular, if (λ1 - λ2)n + k - λ1 is not a square, we prove that all the corresponding divisible difference sets can be constructed by using certain partial difference sets. Also, we determine the structure of reversible divisible difference sets if a Sylow subgroup of G is cyclic. As a consequence, we completely characterize all reversible divisible difference sets in cyclic groups. Finally, the case that -1 is a weak multiplier is studied and restrictions on the parameters are obtained. In fact, we show that n must be a power of 2. © 1992.