Enhanced Power Graphs of Finite Groups

Abstract

The enhanced power graph ${\mathcal{G}}_e(\mathbf{G})$ of a group $\mathbf{G}$ is the graph with vertex set $G$ such that two vertices $x$ and $y$ are adjacent if they are contained in a same cyclic subgroup. We prove that finite groups with isomorphic enhanced power graphs have isomorphic directed power graphs. We show that any isomorphism between power graphs of finite groups is an isomorhism between enhanced power graphs of these groups, and we find all finite groups $\mathbf{G}$ for which $\mathrm{Aut}({\mathcal{G}}_e(\mathbf{G})$ is abelian, all finite groups $\mathbf{G}$ with $|\mathrm{Aut}({\mathcal{G}}_e(\mathbf{G})|$ being prime power, and all finite groups $\mathbf{G}$ with $|\mathrm{Aut}({\mathcal{G}}_e(\mathbf{G})|$ being square free. Also we describe enhanced power graphs of finite abelian groups. Finally, we give a characterization of finite nilpotent groups whose enhanced power graphs are perfect, and we present a sufficient condition for a finite group to have weakly perfect enhanced power graph.