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 (H) over tilde = H \ {1, omega}, where H = G/ N and omega = Pi(sigma is an element of H)sigma. Using D we define a two- to- one map g from (H) over tilde to N. The map g satisfies g(sigma(m)) = g(sigma)(m) and g(sigma) = g(sigma(-1)) for any multiplier m of D and any element sigma is an element of (H) over tilde. As applications, we present some results which give a restriction on the possible order n and the group theoretic structure of G/N.
affine difference set multiplier relative difference set