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Transactions of the American Mathematical Society

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

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On 2-Generator Subgroups of SO(3)
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by Charles Radin and Lorenzo Sadun PDF
Trans. Amer. Math. Soc. 351 (1999), 4469-4480 Request permission

Abstract:

We classify all subgroups of $SO(3)$ that are generated by two elements, each a rotation of finite order, about axes separated by an angle that is a rational multiple of $\pi$. In all cases we give a presentation of the subgroup. In most cases the subgroup is the free product, or the amalgamated free product, of cyclic groups or dihedral groups. The relations between the generators are all simple consequences of standard facts about rotations by $\pi$ and $\pi /2$. Embedded in the subgroups are explicit free groups on 2 generators, as used in the Banach-Tarski paradox.
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Additional Information
  • Charles Radin
  • Affiliation: Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712
  • MR Author ID: 194238
  • Email: radin@math.utexas.edu
  • Lorenzo Sadun
  • Affiliation: Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712
  • Email: sadun@math.utexas.edu
  • Received by editor(s): October 13, 1997
  • Published electronically: June 10, 1999
  • Additional Notes: Research of the first author was supported in part by NSF Grant No. DMS-9531584.
    Research of the second author was supported in part by NSF Grant No. DMS-9626698.
  • © Copyright 1999 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 351 (1999), 4469-4480
  • MSC (1991): Primary 51F25, 52C22
  • DOI: https://doi.org/10.1090/S0002-9947-99-02397-1
  • MathSciNet review: 1624202