The Power Graph of a Torsion-Free Group Determines the Directed Power Graph

TL;DR

This paper proves that for certain torsion-free groups, the structure of the power graph uniquely determines the directed power graph, establishing a strong link between these two graph representations.

Contribution

It demonstrates that, under specific conditions, the power graph fully determines the directed power graph for torsion-free groups, a novel result in group theory graph analysis.

Findings

Power graph determines directed power graph for certain torsion-free groups

Isomorphic power graphs imply isomorphic directed power graphs in these groups

Conditions exclude groups with specific quasicyclic subgroups

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 of $\mathbf G$, denoted with $\mathcal G(\mathbf G)$, is the underlying simple graph. In this paper, for groups $\mathbf G$ and $\mathbf H$, the following is proved. If $\mathbf G$ has no quasicyclic subgroup $\mathbf C_{p^\infty}$ which has trivial intersection with every cyclic subgroup $\mathbf K$ of $\mathbf G$ such that $\mathbf K\not\leq\mathbf C_{p^\infty}$, then $\mathcal G(\mathbf G)\cong \mathcal G(\mathbf H)$ implies $\vec{\mathcal G}(\mathbf G)\cong \vec{\mathcal G}(\mathbf H)$. Consequently, any two torsion-free groups having isomorphic power graphs have isomorphic directed power graphs.