Extension of a theorem of Baayen and Helmberg on monothetic groups

Abstract:

Let $G$ and $K$ be compact monothetic groups and let $\phi$ be a continuous homomorphism from $G$ onto $K$. If $k$ is a generator of $K$, must there exist a generator $g$ of $G$ such that $\phi \left ( g \right ) = k?$? A useful theorem of Baayen and Helmberg provides an affirmative answer if $K$ is the circle $T$. We show that the answer remains affirmative as long as $K$ is metrizable. We also provide an example to show that the answer may be negative for nonmetrizable $K$.