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Products of conjugacy classes and fixed point spaces

Authors: Robert Guralnick and Gunter Malle
Journal: J. Amer. Math. Soc. 25 (2012), 77-121
MSC (2010): Primary 20C15, 20C20, 20D05; Secondary 20E28, 20E45, 20F10, 20F69
Published electronically: June 27, 2011
MathSciNet review: 2833479
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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

Gunter Malle
Affiliation: FB Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany
MR Author ID: 225462

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.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.