Metacommutation as a Group Action on the Projective Line Over $\mathbb {F}_p$
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- by Adam Forsyth, Jacob Gurev and Shakthi Shrima PDF
- Proc. Amer. Math. Soc. 144 (2016), 4583-4590 Request permission
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 $p$.References
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Additional Information
- Adam Forsyth
- Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
- Email: adamforsyth@stanford.edu
- Jacob Gurev
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- Email: gurev@mit.edu
- Shakthi Shrima
- Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08540
- Email: sshrima@princeton.edu
- Received by editor(s): March 18, 2015
- Received by editor(s) in revised form: December 24, 2015
- Published electronically: July 21, 2016
- Communicated by: Ken Ono
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 4583-4590
- MSC (2010): Primary 11R52; Secondary 11R27
- DOI: https://doi.org/10.1090/proc/13126
- MathSciNet review: 3544510