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 ∈ H 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 − 1 } = { ρ , ρ − 1 } and show that ord o ( σ ) ( m ) / ord o ( g ( σ ) ) ( 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’s theorem on even order n [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 ) ([K.T. Arasu, A. Pott, On quasi-regular collineation groups of projective planes, Designs Codes and Cryptography 1 (1991) 83–92] Section 5).