Classification of semiregular relative difference sets with (λ ,n)=1 attaining Turyn’s bound

Suppose a (λ n,n,λ n, λ ) relative difference set exists in an abelian group G=S× H , where |S|=λ , |H|=n^2 , (λ ,n)=1 , and λ is self-conjugate modulo λ n . Then λ is a square, say λ =u^2 , and exp (S) divides u by Turyn’s exponent bound. We classify all such relative difference sets with exp (S)=u . We also show that n must be a prime power if an abelian (λ n, n, λ n, λ ) RDS with (λ ,n)=1 exists and λ is self-conjugate modulo n.