Enhanced Power Graphs of Finite Groups

TL;DR

This paper studies the enhanced power graph of finite groups, establishing isomorphism relations with other power graphs, characterizing automorphism groups, and describing structures for specific classes like abelian and nilpotent groups.

Contribution

It proves that finite groups with isomorphic enhanced power graphs have isomorphic directed power graphs and characterizes automorphism groups of enhanced power graphs for various finite groups.

Findings

Finite groups with isomorphic enhanced power graphs have isomorphic directed power graphs.

Characterization of automorphism groups of enhanced power graphs for finite groups.

Descriptions of enhanced power graphs for finite abelian and nilpotent 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 $\lvert\mathrm{Aut}(\mathcal G_e(\mathbf G)\rvert$ being prime power, and all finite groups $\mathbf G$ with $\lvert\mathrm{Aut}(\mathcal G_e(\mathbf G)\rvert$ 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.