Cocyclic generalised Hadamard matrices and central relative difference sets (Q1273539)

The main result is the equivalence of the existence of: (1) a \(G\)-cocyclic generalized Hadamard matrix GH\((w,v/w)\) with entries in \(C\), (2) a relative \((v,w,v,v/w)\)-difference set in a central extension \(E\) of \(C\) by \(G\) relative to \(C\), and (3) a square divisible \((v,w,v,v/w)\)-design, class regular with respect to \(C\) with a central extension \(E\) of \(C\) by \(G\) as a regular group of automorphisms, where \(G\) is a finite group of order \(v\), and \(C\) is a finite abelian group of order \(w\) such that \(w| v\). This nice theorem generalizes a result of Jungnickel on splitting relative \((v,w,v,v/w)\)-difference sets and a result of Launey on \(G\)-cocyclic Hadamard matrices. In the last section the authors look at constructions and restrictions on the parameters for \(G\)-cocyclic generalized Hadamard matrices.