Undirected power graph need not determine directed power graph for infinite group

Statement

It is possible to have infinite groups $G$ and $H$ such that $G$ and $H$ have isomorphic undirected power graphs but do not have isomorphic directed power graphs. In particular, they need not be 1-isomorphic groups (?).

Proof

Further information: quasicyclic group

For any prime number $p$, the $p$-quasicyclic group is, up to isomorphism, the group of the union of all $(p^{n})^{th}$ roots of unity for all nonnegative integers $n$, under multiplication of complex numbers. Equivalently, it is the direct limit of a sequence of groups of order $p^{n}$ with each group injecting into the next one naturally.

The undirected power graph of any $p$-quasicyclic group is a countable complete graph. Hence, the undirected power graphs of $p$-quasicyclic groups for different primes $p$ are isomorphic. On the other hand, the directed power graphs are not isomorphic, because we can use the directed power graph to determine whether an element has order $p$ (by counting the edges outward from it) and hence use this to distinguish between $p$-quasicyclic groups for different $p$.

References

• The power graph of a finite group, II by Peter J. Cameron, ,  : ^{official online copy} The article has not yet been printed.

^{More info} -- mentioned as a brief aside in this paper.