Suppose a \((\lambda n,n,\lambda n, \lambda )\) relative difference set exists in an abelian group \(G=S\times H\), where \(|S|=\lambda \), \(|H|=n^2\), \(\gcd (\lambda ,n)=1\), and \(\lambda \) is self-conjugate modulo \(\lambda n\). Then \(\lambda \) is a square, say \(\lambda =u^2\), 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 \((\lambda n, n, \lambda n, \lambda )\) RDS with \(\gcd (\lambda ,n)=1\) exists and \(\lambda \) is self-conjugate modulo n.

假设阿贝尔群\(G=S\times H\)中存在一个\((\lambda n,n,\lambda n, \lambda )\)相对差分集，其中\(|S|=\lambda \)、\(|H|=n^2\)、\(\gcd (\lambda ,n)=1\)和\(\lambda \)是自共轭模\(\lambda n\)。那么\(\lambda \)是一个平方，即\(\lambda =u^2\)，并且\(\exp (S)\)将u除以 Turyn 的指数界。我们用\(\exp (S)=u\)对所有此类相对差异集进行分类。我们还证明，如果存在具有\ (\gcd (\lambda ,n)=1\) 的阿贝尔\((\lambda n, n, \lambda n, \lambda )\) RDS 且\ (\lambda \)是自共轭模n。