Countable number of subgroups $\implies $ countable group

Question

I know that if a group $G$ has a finite number of subgroups then the group $G$ is finite.

But if a group $G$ has countable number of subgroups then is the group countable?

Answer 1

Let $S$ be the set of all subgroups of $G$.

Consider the map $\phi: G \to S$ given $\phi(g) = \langle g \rangle$.

If $\phi$ were injective, then we'd be done.

Unfortunately, $\phi$ is not injective, but fortunately we can control how not injective it is.

Indeed, every $H \in S$ has a finite number of pre-images since every cyclic group has finite number of generators (including infinite cyclic groups). Therefore $$ G = \bigcup_{H \in S} \phi^{-1}(H) $$ is a countable union of finite sets and so is countable.