Products of conjugacy classes and fixed point spaces

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 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 r th powers for any prime power r r (in particular, a product of two squares answering a conjecture of Larsen, Shalev and Tiep).