Simple groups admit Beauville structures

We prove a conjecture of Bauer, Catanese and Grunewald showing that all finite simple groups other than the alternating group of degree 5 admit unmixed Beauville structures. We also consider an analog of the result for simple algebraic groups which depends on some upper bounds for character values of regular semisimple elements in finite groups of Lie type. Finally, we prove that any finite simple group contains two conjugacy classes C and D such that any pair of elements in C × D generates the group.