Products of conjugacy classes and fixed point spaces

Guralnick, Robert¹ ; Malle, Gunter²

¹ Department of Mathematics, University of Southern California, 3620 S. Vermont Avenue, Los Angeles, California 90089-2532
² FB Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany

Journal of the American Mathematical Society, Tome 25 (2012) no. 1, pp. 77-121

Résumé

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).

Détail

DOI : 10.1090/S0894-0347-2011-00709-1

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