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

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 $x\to y$ 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 $\mathbf{K}\nleqq {\mathbf{C}}_{p^{\infty }}$, 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.