On affine difference sets and their multipliers

Let DD be an affine difference set of order nn in an abelian group GG relative to a subgroup NN. We denote by π(s)π(s) the set of primes dividing an integer s(>0) and set H∗=H∖{ω}H∗=H∖{ω}, where H=G/NH=G/N and ω=∏σ∈Hσω=∏σ∈Hσ. In this article, using DD we define a map gg from HH to NN satisfying for τ,ρ∈H∗,g(τ)=g(ρ) iff {τ,τ−1}={ρ,ρ−1}{τ,τ−1}={ρ,ρ−1} and show that ordo(σ)(m)/ordo(g(σ))(m)∈{1,2} for any σ∈H∗σ∈H∗ and any integer m>0m>0 with π(m)⊂π(n)π(m)⊂π(n). This result is a generalization of J.C. Galati’s theorem on even order nn [J.C. Galati, A group extensions approach to affine relative difference sets of even order, Discrete Mathematics 306 (2006) 42–51] and gives a new proof of a result of Arasu–Pott on the order of a multiplier modulo exp(H)(H) ([K.T. Arasu, A. Pott, On quasi-regular collineation groups of projective planes, Designs Codes and Cryptography 1 (1991) 83–92] Section 5).