Products of conjugacy classes in finite and algebraic simple groups

We prove the Arad–Herzog conjecture for various families of finite simple groups — if AA and BB are nontrivial conjugacy classes, then ABAB is not a conjugacy class. We also prove that if GG is a finite simple group of Lie type and AA and BB are nontrivial conjugacy classes, either both semisimple or both unipotent, then ABAB is not a conjugacy class. We also prove a strong version of the Arad–Herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes. As a consequence we obtain a complete classification of pairs of centralizers in a simple algebraic group which have dense product. A special case of this has been used by Prasad to prove a uniqueness result for Tits systems in quasi-reductive groups. Our final result is a generalization of the Baer–Suzuki theorem for pp-elements with p≥5p≥5.