In this paper we study finite abelian groups admitting a d~fference set with multiplier -1. In these groups we have that each integer, which is relatively prime to the group order, is a multiplier (see [1] and Section I of this paper).About the arithmetical structure, there is an interesting result of Jungnickel 1-3] on primes dividing the order n of the corresponding design. Here we prove (see Theorem 2.1) that each odd prime divisor of the order v of the group divides n. The proof of Theorem 2.1 rests on character computations and is motivated by [5].

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