On the non-existence of generalized Hadamard matrices (Q5896511)

For a generalized Hadamard matrix H of order n over G, the author considers its determinant over the integral group ring \({\mathbb{R}}(G)\) factored by \({\mathbb{Z}}(\sum_{g\in G}g)\). It is shown that \(\det (H)\det (H^*)=n^ n;\) the number-theoretic consequences of this are explored, showing the non-existence of many generalized Hadamard matrices for groups whose orders are divisible by 3, 5 or 7. Also, for some n and G, the possible values for det(H) are found.