Directed power graph-equivalent not implies 1-isomorphic for infinite groups

Statement
It is possible to have two infinite groups $$G$$ and $$H$$ that are directed power graph-equivalent (i.e., they have isomorphic directed power graphs) and are not fact about::1-isomorphic groups.

Related facts

 * Finite groups are 1-isomorphic iff their directed power graphs are isomorphic
 * Undirected power graph determines directed power graph for finite group

Proof
For any prime $$p$$, consider the group $$G_p$$ of rational numbers that, in reduced form, have denominator a power of $$p$$. The isomorphism class of the directed power graph of $$G_p$$ is independent of $$p$$.