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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles 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.

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Nonabelian free subgroups in homomorphic images of valued quaternion division algebras
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by Andrei S. Rapinchuk, Louis Rowen and Yoav Segev PDF
Proc. Amer. Math. Soc. 134 (2006), 3107-3114 Request permission


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.
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Additional Information
  • Andrei S. Rapinchuk
  • Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
  • MR Author ID: 206801
  • Email:
  • Louis Rowen
  • Affiliation: Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel
  • MR Author ID: 151270
  • Email:
  • Yoav Segev
  • Affiliation: Department of Mathematics, Ben-Gurion University, Beer-Sheva 84105, Israel
  • MR Author ID: 225088
  • Email:
  • 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:
  • MathSciNet review: 2231891