On Hadamard difference sets with weak multiplier minus one (Q1409282)

An Hadamard (or Menon) difference set is a \((v,k,\lambda)\)-design such that \(v=4(k-\lambda)\). The rational integer \(t\) is called a weak multiplier of the difference set \(D\) of a group \(G\) if \(D^t:=\{x^t\mid x\in D\}=Dg\) for some \(g\in G\). Let \(G\) be the direct product of \(\mathbb{Z}_4\) and a group \(H\) of odd order \(u^2\). If \(G\) contains an Hadamard difference set with weak multiplier \(-1\) then \(u\) divides the order of the derived \(H'(=[H,H])\) of \(H\). Proofs employ the group ring \(\mathbb{Z} G\) and the characters of \(G\); the more general case where the product is semidirect is studied.