Classifying topologically cyclic groups

Question

The group of p-adic integers has a dense cyclic subgroup, i.e. it is topologically cyclic

What are some (or all) other non-trivial (i.e. non-discrete) examples of such groups?

Answer 1

Such groups are also called monothetic and there are a lot of them. For instance, here is an abstract of a paper “Subgroups of monothetic groups” by Sidney A. Morris and Vladimir Pestov:

“It is shown that every separable abelian topological group is isomorphic with a topological subgroup of a monothetic group (that is, a topological group with a single topological generator). In particular, every separable metrizable abelian group embeds into a metrizable monothetic group. More generally, we describe all topological groups that can be embedded into monothetic groups: they are exactly abelian topological groups of weight $\le \frak c$ covered by countably many translations of every nonempty open subset”.

Such $T_0$ groups should be exactly subgroups of products of not greater than $\frak c$ second countable (that is. separable and metrizable) abelian topological groups.

From the other side, for a locally compact ($T_0$) monothetic topological group $G$ there is a Pontrjagin alternative: $G$ is discrete or $G$ is compact.

Also it seems the following.

Example. Let $\Bbb T=\{z\in\Bbb C:|z|=1\}$ be the unit circle endowed with the standard topology. Let $\{g_\alpha:\alpha<\frak c\}$ be a linearly independent subset of the group $\Bbb T$, that is, for each finite subset $F$ of $\frak c$ and each map $n:F\to\Bbb Z$ an equality

$$\sum_{\alpha\in F} n(\alpha)g_\alpha=0$$

implies $n(\alpha)=0$ for each $\alpha\in F$.

Let $\lambda\le\frak c$ be an arbitrary cardinal. Put $G={\Bbb T}^\lambda$. Then $G$ is a (compact by Tychonov Theorem) topological group. Define an element $\overline{g}=(\overline{g}_\alpha)_{ \alpha<\lambda}\in G$ such that $\overline{g}_\alpha=g_\alpha $ for each $\alpha<\lambda$. Example 65 from [Pon] implies that the cyclic group generated by the element $\overline{g}$ is dense in $G$, so the group $G$ is monothetic.

References

[Pon] Lev S. Pontrjagin, Continuous groups, 2nd ed., M., (1954) (in Russian).