Classification of finite simple groups and p-groups with a special property on their conjugacy classes

Let G be a finite group and let x^G denotes the conjugacy class of an element x of G . We say that group G satisfies property Ui , i ≥1, if union of any i distinct non-trivial conjugacy classes of G together with 1 is a subgroup of it. In this paper, we classify all finite simple groups G that satisfy property U5 and we prove that there is no finite p - group satisfying property U5 .