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Journal of the American Mathematical Society

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

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Products of conjugacy classes and fixed point spaces
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by Robert Guralnick and Gunter Malle PDF
J. Amer. Math. Soc. 25 (2012), 77-121 Request permission

Abstract:

We prove several results on products of conjugacy classes in finite simple groups. The first result is that for any finite nonabelian simple groups, there exists a triple of conjugate elements with product $1$ which generate the group. This result and other ideas are used to solve a 1966 conjecture of Peter Neumann about the existence of elements in an irreducible linear group with small fixed space. We also show that there always exist two conjugacy classes in a finite nonabelian simple group whose product contains every nontrivial element of the group. We use this to show that every element in a nonabelian finite simple group can be written as a product of two $r$th powers for any prime power $r$ (in particular, a product of two squares answering a conjecture of Larsen, Shalev and Tiep).
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Additional Information
  • Robert Guralnick
  • Affiliation: Department of Mathematics, University of Southern California, 3620 S. Vermont Avenue, Los Angeles, California 90089-2532
  • MR Author ID: 78455
  • Email: guralnic@usc.edu
  • Gunter Malle
  • Affiliation: FB Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany
  • MR Author ID: 225462
  • Email: malle@mathematik.uni-kl.de
  • Received by editor(s): May 20, 2010
  • Received by editor(s) in revised form: June 4, 2010, January 11, 2011, and April 19, 2011
  • Published electronically: June 27, 2011
  • Additional Notes: The first author was partially supported by NSF grants DMS 0653873 and 1001962.
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 25 (2012), 77-121
  • MSC (2010): Primary 20C15, 20C20, 20D05; Secondary 20E28, 20E45, 20F10, 20F69
  • DOI: https://doi.org/10.1090/S0894-0347-2011-00709-1
  • MathSciNet review: 2833479