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

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