Cohen’s Theorem and \(Z\)-Cyclic Whist Tournaments

Let \(p,q\) denote primes, \(p \equiv 1 \pmod{4}\), \(g \equiv 3 \pmod{4}\), \(g \geq 7\). In an earlier study we established that if \(\gcd(q-1, p^{n-1}(p-1)) = 2\) and if a \(\mathbb{Z}\)-cyclic \(Wh(q+1)\) exists then a \(\mathbb{Z}\)-cyclic \(Wh(qp^n + 1)\) exists for all \(n \geq 0\). Here we consider \(\gcd(qg-1,p^{n-1}(p-1)) > 2\) and prove that if a \(\mathbb{Z}\)-cyclic \(Wh(q+1)\) exists then there exists a \(\mathbb{Z}\)-cyclic \(Wh(qp^n + 1)\) for all \(n \geq 0\). The proof employed depends on the existence of an appropriate primitive root of \(p\). Utilizing a theorem of S. D. Cohen we establish that such appropriate primitive roots always exist.