Finite groups whose order graph is C4-free

Given a finite group G , the order graph of G, denoted by S(G), is a graph whose vertex set is G, and two distinct vertices a and b are adjacent if o(a) | o(b) or o(b) | o(a), where o(a), and o(b), are the orders of a and b in G, respectively. In this paper, by the order of an element, we give a characterization of the finite groups whose order graph is C4-free. As applications, we classify a few families of finite groups whose order graph is C4-free, such as nilpotent groups, dihedral groups and symmetric groups.