Abstract:

This paper studies properties of power graphs of nilpotent groups, inner nilpotent groups and inner abelian groups. In general, a power graph ${\mathscr{P}}(G)$ of a finite group $G$ is a line graph if and only if $G$ is a cyclic group of a prime power order. In addition, the necessary and sufficient conditions of power graphs for independent numbers of nilpotent groups and inner abelian groups that take the critical value and the planarization of inner nilpotent groups and inner abelian groups are obtained. Finally, the connectivity of proper power graphs of inner nilpotent groups and inner abelian groups are discussed. This paper gets the diameter estimations and the numbers of connected components under the conditions of connected and disconnected proper power graphs, respectively.

Key words: nilpotent group, inner nilpotent group, inner abelian group, power graph