Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Nonabelian free subgroups in homomorphic images of valued quaternion division algebras
HTML articles powered by AMS MathViewer

by Andrei S. Rapinchuk, Louis Rowen and Yoav Segev PDF
Proc. Amer. Math. Soc. 134 (2006), 3107-3114 Request permission

Abstract:

Given a quaternion division algebra $D,$ a noncentral element $e \in D^\times$ is called pure if its square belongs to the center. A theorem of Rowen and Segev (2004) asserts that for any quaternion division algebra $D$ of positive characteristic $> 2$ and any pure element $e \in D^\times$ the quotient $D^{\times }/X(e)$ of $D^{\times }$ by the normal subgroup $X(e)$ generated by $e,$ is abelian-by-nilpotent-by-abelian. In this note we construct a quaternion division algebra $D$ of characteristic zero containing a pure element $e\in D$ such that $D^\times /X(e)$ contains a nonabelian free group. This demonstrates that the situation in characteristic zero is very different.
References
Similar Articles
Additional Information
  • Andrei S. Rapinchuk
  • Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
  • MR Author ID: 206801
  • Email: asr3x@unix.mail.virginia.edu
  • Louis Rowen
  • Affiliation: Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel
  • MR Author ID: 151270
  • Email: rowen@macs.biu.ac.il
  • Yoav Segev
  • Affiliation: Department of Mathematics, Ben-Gurion University, Beer-Sheva 84105, Israel
  • MR Author ID: 225088
  • Email: yoavs@math.bgu.ac.il
  • Received by editor(s): March 3, 2005
  • Received by editor(s) in revised form: May 14, 2005
  • Published electronically: May 11, 2006
  • Additional Notes: The first author was partially supported by BSF grant 2000-171, and by NSF grants DMS-0138315 and DMS-0502120.
    The second author was partially supported by the Israel Science Foundation Center of Excellence.
    The third author was partially supported by BSF grant 2000-171.
  • Communicated by: Jonathan I. Hall
  • © Copyright 2006 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 3107-3114
  • MSC (2000): Primary 16K20, 16U60; Secondary 20G15, 12J20
  • DOI: https://doi.org/10.1090/S0002-9939-06-08385-7
  • MathSciNet review: 2231891