Proper connection of power graphs of finite groups

Let G be a finite group. The power graph of G is the undirected graph whose vertex set is G, and two distinct vertices are adjacent if one is a power of the other. The reduced power graph of G is the subgraph of the power graph of G obtained by deleting all edges {x, y} with < x > = < y >, where x and y are two distinct elements of G. In this paper, we determine the proper connection number of the reduced power graph of G. As an application, we also determine the proper connection number of the power graph of G.