On the independence number of the power graph of a finite group

The power graph ΓG of a finite group G is the graph whose vertex set is G, two distinct elements being adjacent if one is a power of the other. In this paper, we give sharp lower and upper bounds for the independence number of ΓG and characterize the groups achieving the bounds. Moreover, we determine the independence number of ΓG if G is cyclic, dihedral or generalized quaternion. Finally, we classify all finite groups G whose power graphs have independence number 3 or n−2, where n is the order of G.