Some non-existence results on divisible difference sets (Q1180404)

Many interesting results are obtained using characters and multipliers in conjunction with ordinary difference sets. We can only quote the following sample theorems. If the abelian group \(G\) has a \((m,n,k,\lambda_ 1,\lambda_ 2)\)-DDS and \(m=4u+2\) then \(k-\lambda_ 1\) is a perfect square or the 2-Sylow subgroup of \(G\) is cyclic. Let the group \(G\) be the direct sum of \(N\) (abelian) and the group \(H\). If the order of \(N\) is even and \(G\) has a \((m,n,k,\lambda_ 1,\lambda_ 2)\)- DDS then \(k-\lambda_ 1\) is a perfect square. If \(G\) has a \((n,n,n,1)\)- RDS then each odd prime \(p\) dividing the square-free part of \(n\) is a quadratic residue modulo \(q\) for every prime divisor \(q\neq p\) of \(n\).