A two-to-one map and abelian affine difference sets

Abstract

Let D be an affine difference set of order n in an abelian group G relative to a subgroup N. Set ${\tilde{H}}$ = H \ {1, ω}, where H = G/N and ${\omega=\prod_{\sigma\in H}\sigma}$ . Using D we define a two-to-one map g from ${\tilde{H}}$ to N. The map g satisfies g(σ ^m) = g(σ)^m and g(σ) = g(σ ⁻¹) for any multiplier m of D and any element σ ∈ ${\tilde{H}}$ . As applications, we present some results which give a restriction on the possible order n and the group theoretic structure of G/N.