Products of conjugacy classes in simple groups

Let G be a finite group. For a ∈ G, let a ^G = {a ^g {pipe} g ∈ G} be the conjugacy class of a in G. In this paper, we study a conjecture due to Arad and Herzog which asserts that in a finite non-abelian simple group the product of two nontrivial conjugacy classes is never a single conjugacy class. In particular, we will verify this conjecture for several families of finite simple groups of Lie type.