Multiplier groups of difference sets

AbstractIn this paper, we consider (v, k, λ)-difference sets from the point of view of their multiplier groups. Let ζ be a primitive v-th root of unity, Q, the field of rational numbers and Q(ζ), the field of v-th roots of unity. We apply the Galois theory to the field Q(ζ) and its Galois group over Q. Since the multiplier group of a difference set modulo v is isomorphic to a subgroup of this Galois group, this approach yields some characterisations of the difference set. We prove the following main results: Theorem. Let t be a multiplier of a (v, k, λ)-difference set D. Let q be a prime divisor of v and ω, a primitive q-th root of unity. Set θ = ωd1 + … + ωdk where d1,…, dk is a translate of D which is fixed by t and let r be the degree of θ over Q. Then t is an r-th power residue modulo q. Theorem: Let D be a (v, k, λ)-difference set and let D be a multiplicative group modulo v. If D has an integer t such that (t − 1, v) = 1, then the multiplier group of D coincides with D.As applications of the above theorems, we derive two results of Emma Lehmer on Residue Difference Sets and a result of H. B. Mann