On the Cycle Structure of the Metacommutation Map

Abstract

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 $ξ$ of norm $q$ has either $0$, $1$ or $2$ fixed points, and that the permutation $τ_{ξ,p}$ 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 $ξ$, using cyclotomic polynomials over $\mathbb{F}_p$. We then show that, given an odd prime $p$, there is always a prime quaternion $ξ$ such that the permutation $τ_{ξ,p}$ has only one non-trivial cycle of length $p$. Finally, we give conditions for a prime $π$ of norm $p$ to be a fixed point of the aforementioned metacommutation map.