On affine difference sets and their multipliers

Let D be an affine difference set of order n in an abelian group G relative to a subgroup N. We denote by π(s) the set of primes dividing an integer s(>0) and set H* = H∖{ω}, where H = G/N and ω=∏_σ. In this article, using D we define a map g from H to N satisfying for τ, ρ∈H*, g(τ)=g(ρ) iff {τ,τ^}={ρ,ρ^} and show that ord_(m)/ord_(m)∈{1,2} for any σ∈H* and any integer m>0 with π(m)⊂π(n). This result is a generalization of J.C. Galati\u27s theorem on even order n([3]) and gives a new proof of a result of Arasu–Pott on the order of a multiplier modulo exp(H) ([1]Section 5)