Determine whether $(\Bbb R,+,0)$ is finitely generated

Question

I have to determine whether $(\Bbb R,+,0)$ is finitely generated. I am thinking of considering the following: if the group is finite Abelian then it is finitely generated. Would it suffice to show that it is abelian but it is not finite?

Answer 1

There are two questions:

• If $A$ and $B$ implies $C$, and if $A$ but not $B$, then not $C$?

Obviously, no.

• Is the additive group of the reals finitely-generated?

No. Let $S$ be a finite subset in $\mathbb{R}$. Let $\phi : \mathbb{Z}^S \rightarrow \mathbb{R}$ be the map sending $(n_s)_{s \in S}$ to $\sum_{s \in S} n_s s$, then $\phi$ is a group morphism. Since $\mathbb{Z}^S$ is countable, and $\mathbb{R}$ is not, it cannot be surjective.