On finite groups whose power graph is a cograph

A $P_4$-free graph is called a cograph. In this paper we partially characterize finite groups whose power As will see, problem generalization of the determination in which every element has prime order, first raised by Graham Higman 1957 and fully solved very recently. First determine all $G$ $H$ for $G\times H$ We show that cograph can be characterised condition only involving elements orders are or product two (possibly equal) primes. Some important classes also taken under consideration. For simple most cases their graphs not cographs: ones cographs certain PSL$(2,q)$ Sz$(q)$ group PSL$(3,4)$. However, complete these involves some hard number-theoretic problems.