Structure of group invariant weighing matrices of small weight

We show that every weighing matrix of weight n invariant under a finite abelian group G can be generated from a subgroup H of G with |H|≤2n−1. Furthermore, if n is an odd prime power and a proper circulant weighing matrix of weight n and order v exists, then v≤2n−1. We also obtain a lower bound on the weight of group invariant matrices depending on the invariant factors of the underlying group. These results are obtained by investigating the structure of subsets of finite abelian groups that do not have unique differences.