Power graphs of all nilpotent groups

TL;DR

This paper characterizes nilpotent groups based on their power graphs, showing the Pr"ufer group as unique in not being fully determined by its power graph, and identifies groups uniquely determined by their power graphs.

Contribution

It proves that the Pr"ufer group is the only nilpotent group with a power graph that does not determine its directed power graph, and identifies a class of groups uniquely determined by their power graphs.

Findings

Pr"ufer group is uniquely characterized among nilpotent groups by its power graph.

A specific group with quasicyclic torsion subgroup is determined by its power graph.

Power graphs can distinguish certain nilpotent groups up to isomorphism.

Abstract

The directed power graph $\vec{\mathcal G}(\mathbf G)$ of a group $\mathbf G$ is the simple digraph with vertex set $G$ such that $x\rightarrow y$ if $y$ is a power of $x$. The power graph $\mathcal G(\mathbf G)$ of the group $\mathbf G$ is the underlying simple graph. In this paper, we prove that Pr\"ufer group is the only nilpotent group whose power graph does not determine the directed power graph up to isomorphism. Also, we present a group $\mathbf G$ with quasicyclic torsion subgroup that is determined by its power graph up to isomorphism, i.e. such that $\mathcal G(\mathbf H)\cong\mathcal G(\mathbf G)$ implies $\mathbf H\cong \mathbf G$ for any group $\mathbf H$.