On the Cycle Structure of the Metacommutation Map

Cohn and Kumar showed that the permutation on the set of the classes of left associated Hurwitz primes above an odd prime $p$ induced through metacommutation by a Hurwitz prime $\xi$ of norm $q$ has either $0$, $1$ or $2$ fixed points, and that the permutation $\tau_{\xi,p}$ induced on the non-fixed points splits into cycles of the same length. Here we show how to find the length of those cycles, in terms of $p$ and $\xi$, using cyclotomic polynomials over $\mathbb{F}_p$. We then show that, given an odd prime $p$, there is always a prime quaternion $\xi$ such that the permutation $\tau_{\xi,p}$ has only one non-trivial cycle of length $p$. Finally, we give conditions for a prime $\pi$ of norm $p$ to be a fixed point of the aforementioned metacommutation map.