Compact subgroups of a topological group

Question

Consider $G=(0,\infty )$ , with the metric induced by Euclidean metric from $\mathbb R$ .$G$ is a group under multiplication . Then which subgroups of it are compact $?$

Now $G=\mathbb R_+$ and there are subgroups $\mathbb Z_+$,$\mathbb Q_+$, $\mathbb Q^c_+$ ; none of which is compact under Euclidean metric .

The compact subsets of $\mathbb R_+$ are :

• Closed intervals $[a,b]$- which are not subgroups.
• Finite sets $\{a_1,a_2,...,a_n\}$;$n\gt 1$ - which are also not subgroups.
• Singletons $\{b\}$ - which are also not subgroups except $\{1\}$ .

So $\{1\}$ is the only subgroup that is compact .

Is this correct or are there more $?$

Thanks for any help.

Answer 1

There are no non-trivial compact subrgroups of the group $G$, because each non-unit element $g\in G$ generates an unbounded subgroup $\langle g\rangle=\{g^n:n\in\Bbb Z\}$ which cannot be contained in a compact subset of the space $G$.