  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 $\mathbf H$.
