The Power Graph of a Torsion-Free Group of Nilpotency Class $2

The directed power graph $\mathcal G(\mathbf G)$ of a group $\mathbf G$ is the simple digraph with vertex set $G$ in which $x\rightarrow y$ if $y$ is a power of $x$, the power graph is the underlying simple graph, and the enhanced power graph of $\mathbf G$ is the simple graph with the same vertex set such that two vertices are adjacent if they are powers of some element of $\mathbf G$. In this paper three versions of the definition of the power graphs are discussed, and it is proved that the power graph by any of the three versions of the definitions determines the other two up to isomorphism. It is also proved that, if $\mathbf G$ is a torsion-free group of nilpotency class $2$ and if $\mathbf H$ is a group such that $\mathcal G(\mathbf H)\cong\mathcal G(\mathbf G)$, then $\mathbf G$ and $\mathbf H$ have isomorphic directed power graphs, which was an open problem proposed by Cameron, Guerra and Jurina.