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

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