  We prove several results on products of conjugacy classes in finite simple
groups. The first result is that there always exists a uniform generating
triple. 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 non-abelian simple group whose product contains every
nontrivial element of the group. We use this to show that every element in a
non-abelian finite simple group can be written as a product of two rth powers
for any prime power r (in particular, a product of two squares).
