Relation between the power graph of finite group and commutative elements

Let G be a finite group.## The power graph of a group G, with notation P(G) is a graph such that it’s vertex set is the group G and two distinct elements x, y are adjacent if and only if x = yn or y = xn for some positive integer n.## For a nonempety set X of G, The commuting graph C(G, X) is the graph with X as the vertex set and two distinct elements of X being joined by an edge if they are commuting elements of G. The purpose of this paper is study of groups with property P(G) = C(G, G).