1.63 (1965)

A group $G$ is called dense if it has no proper isolated subgroups other than its trivial subgroup.
$\qquad$ a) Do there exist dense torsion-free groups that are not locally cyclic?
$\qquad$ b) Suppose that any two non-trivial elements $x$ and $y$ of a torsion-free group $G$ satisfy the relation $x^k = y^l$, where $k$ and $l$ are non-zero integers depending on $x$ and $y$. Does it follow that $G$ is abelian?