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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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Word maps and Waring type problems
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by Michael Larsen and Aner Shalev PDF
J. Amer. Math. Soc. 22 (2009), 437-466 Request permission


Waring’s classical problem deals with expressing every natural number as a sum of $g(k)$ $k$th powers. Recently there has been considerable interest in similar questions for nonabelian groups and simple groups in particular. Here the $k$th power word is replaced by an arbitrary group word $w \ne 1$, and the goal is to express group elements as short products of values of $w$.

We give a best possible and somewhat surprising solution for this Waring type problem for various finite simple groups, showing that a product of length two suffices to express all elements. We also show that the set of values of $w$ is very large, improving various results obtained so far.

Along the way we also obtain new results of independent interest on character values and class squares in symmetric groups.

Our methods involve algebraic geometry, representation theory, probabilistic arguments, as well as results from analytic number theory, including three primes theorems (approximating Goldbach’s Conjecture).

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Additional Information
  • Michael Larsen
  • Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
  • MR Author ID: 293592
  • Email:
  • Aner Shalev
  • Affiliation: Einstein Institute of Mathematics, Hebrew University, Givat Ram, Jerusalem 91904, Israel
  • MR Author ID: 228986
  • ORCID: 0000-0001-9428-2958
  • Email:
  • Received by editor(s): February 1, 2007
  • Published electronically: September 12, 2008
  • Additional Notes: The first author was partially supported by NSF grant DMS-0354772
    The second author was partially supported by an Israel Science Foundation Grant.
    Both authors were partially supported by a Bi-National Science Foundation United States-Israel Grant.
  • © Copyright 2008 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 22 (2009), 437-466
  • MSC (2000): Primary 20D06, 20G40; Secondary 14G15
  • DOI:
  • MathSciNet review: 2476780