For any integer \(m \geq 2\), let \(\mu_m\) be the group of \(m\)th roots of unity. Let \(p\) be a prime and \(a\) a positive integer. For \(m = p^\alpha\), it is shown that there is no \(n \times n\) matrix over \(\mu_m\) with vanishing permanent if \(n < p\).