Products of conjugacy classes and fixed point spaces

Guralnick, Robert; Malle, Gunter

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

Products of conjugacy classes and fixed point spaces
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by Robert Guralnick and Gunter Malle

J. Amer. Math. Soc. 25 (2012), 77-121

DOI: https://doi.org/10.1090/S0894-0347-2011-00709-1

Published electronically: June 27, 2011

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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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