Note On \((m, 2, m-1, \frac{m-2}{2})\) Relative Difference Sets

Abstract

In this note, we consider relative difference sets with the parameter \((m, 2, m-1, \frac{m-2}{2})\) in a group \(G\) relative to a subgroup \(N\). In the splitting case, \(G = H \times N\), we give a lower bound for the size of the commutator group \(H’\), and we show that \(H\) cannot have a homomorphic image which is generalized dihedral. In the non-splitting case, we prove that there is no \((2n, 2, 2n-1, n-1)\) relative difference set in a generalized dihedral group of order \(4n\), \(n > 1\).