Let G be a finite group. The power graph of G, denoted by P(G), is a graph whose vertex set is G, and two vertices in P(G) are connected by an edge if and only if one of them is a power of another. We start by going over some properties of power graphs. Then we characterize finite groups with no induced subgraphs isomorphic to a path on four vertices. Keywords: Power graph, induced subgraph, cograph, nilpotent group.