Metacommutation as a Group Action on the Projective Line over F_p

2015/03/20 by Adam Forsyth, Forsyth, Adam, Jacob Gurev +3
Mathematics · #11R52 #Advanced Topics in Algebra #Algebraic and Geometric Analysis #Algebraic structures and combinatorial models #FOS: Mathematics #Mathematical Analysis and Transform Methods #Number Theory (math.NT) #math.NT #msc:11R52

paper · pdf · doi:10.48550/arxiv.1503.06259

openalex publication_date 2015/03/20 · arxiv created 2015/03/21 · arxiv updated 2015/03/24 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

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 PGL₂ (\mathbbF_p) on ℙ¹ (\mathbbF_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.

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