Metacommutation as a Group Action on the Projective Line over $\mathbf{F}_\mathbf{p}$

TL;DR

This paper connects metacommutation permutations of Hurwitz primes to the action of PGL(2, F_p) on the projective line, simplifying proofs and characterizing cycle structures, with potential for broader applications.

Contribution

It establishes an equivalence between metacommutation permutations and PGL(2, F_p) actions, providing new insights and simpler proofs for existing results.

Findings

Permutations correspond to PGL(2, F_p) actions on projective lines

Cycle structures of permutations are characterized

Methods extend to all quaternion orders with a division algorithm

Abstract

Cohn and Kumar showed the quadratic character of $q$ modulo $p$ gives the sign of the permutation of Hurwitz primes of norm $p$ induced by the Hurwitz primes of norm $q$ under metacommutation. We demonstrate that these permutations are equivalent to those induced by the right standard action of $\operatorname{PGL}_2 (\mathbb{F}_p)$ on $\mathbb{P}^1 (\mathbb{F}_p)$. This equivalence provides simpler proofs of the results of Cohn and Kumar and characterizes the cycle structure of the aforementioned permutations. Our methods are general enough to extend to all orders over the quaternions with a division algorithm for primes of a given norm $p$.