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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Products of squares in finite simple groups


Authors: Martin W. Liebeck, E. A. O’Brien, Aner Shalev and Pham Huu Tiep
Journal: Proc. Amer. Math. Soc. 140 (2012), 21-33
MSC (2010): Primary 20C33, 20D06
Published electronically: May 6, 2011
MathSciNet review: 2833514
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Abstract: The Ore conjecture, proved by the authors, states that every element of every finite non-abelian simple group is a commutator. In this paper we use similar methods to prove that every element of every finite simple group is a product of two squares. This can be viewed as a non-commutative analogue of Lagrange's four squares theorem. Results for higher powers are also obtained.


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Additional Information

Martin W. Liebeck
Affiliation: Department of Mathematics, Imperial College, Queen’s Gate, London SW7 2BZ, United Kingdom
Email: m.liebeck@imperial.ac.uk

E. A. O’Brien
Affiliation: Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand
Email: e.obrien@auckland.ac.nz

Aner Shalev
Affiliation: Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
Email: shalev@math.huji.ac.il

Pham Huu Tiep
Affiliation: Department of Mathematics, University of Arizona, 617 N. Santa Rita Avenue, Tucson, Arizona 85721
Email: tiep@math.arizona.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10878-5
PII: S 0002-9939(2011)10878-5
Received by editor(s): May 21, 2010
Received by editor(s) in revised form: November 1, 2010
Published electronically: May 6, 2011
Additional Notes: The first author acknowledges the support of a Maclaurin Fellowship from the New Zealand Institute of Mathematics and its Applications
The second author acknowledges the support of the Marsden Fund of New Zealand (grant UOA 0721)
The third author acknowledges the support of ERC Advanced Grant 247034, an EPSRC Visiting Fellowship, an Israel Science Foundation Grant, and Bi-National Science Foundation grant United States-Israel 2008194.
The fourth author acknowledges the support of the NSF (grant DMS-0901241)
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.