Conjugation invariants determine the metacommutation permutation only up to relabelling

Abstract

Let $\mathcal{H}$ be the Hurwitz quaternions, $p$ an odd prime, and $Q \in \mathcal{H}$ a prime of norm $q \neq p$. Metacommutation $PQ = Q'P'$ induces a permutation $π_Q$ of the $p+1$ left-associate classes of primes of norm $p$. Cohn and Kumar compute its sign and fixed-point count, and Leite and Machiavelo its full cycle structure, by formulas depending only on conjugation-invariant data of $Q$ (namely $q$ and $\mathrm{tr}\,Q$). We prove this is exactly the boundary of what such invariants can carry: no quantity $I(Q)$ invariant under unit conjugation determines $π_Q$ as a labelled permutation of the intrinsic class set, or even the image of a single specified class. The proof combines an equivariance identity $π_{uQu^{-1}} = ρ_u π_Q ρ_u^{-1}$ with a minimal, fully explicit witness at $(p,q) = (3,5)$: the four primes $2+i$, $2+j$, $2+k$, $2-i$ form a single unit-conjugacy orbit, hence agree under every conjugation-invariant function, yet induce four pairwise distinct $4$-cycles of the same four classes. We further observe that isomorphisms $\mathcal{H}/p\mathcal{H} \to M_2(\mathbb{F}_p)$ form a torsor under $\mathrm{PGL}_2(\mathbb{F}_p)$, so no projective labelling of the classes is canonical, and that by orbit-stabilizer the datum of one destination $π_Q(C)$ is exactly a coset $g_Q G_C$ in $\mathrm{PGL}_2(\mathbb{F}_p)/G_C$. All sixteen refactorizations in the witness are listed in the appendix and have been verified by machine along two independent routes.