Some results on the power graphs of finite groups

Abstract

We classify planar graphs and complete power graphs of groups and show that the only infinite group with a complete power graph is the Prüfer group ℤp∞. Clique and chromatic numbers and the automorphism group of power graphs are investigated. We also prove that the reduced power graph of a group G is regular if and only if G is a cyclic p-group or exp(G) = p for some prime number p.