Is the following statement true? I think it's a characterization of finite groups.
A group $G$ is finite if and only if every subgroup generated by some $g \in G$ is finite.
The "only if" is immediate, but I can't convince myself of the "if" and am therefore unsure about the claim. The book I'm reading is of the math for physics variety and so doesn't spell this out, though I suspect it might be true.
As I think about it more, perhaps I can argue by the contrapositive? If $G$ is infinite then there must exist an infinite subgroup generated by some $g$ for, if not, then $G \subset \bigcup C_g$ where $C_g$ is the subgroup generated by $g$. But this is absurd since $\bigcup C_g$ is a finite group. Does this reasoning even make sense, and is there a clearer way to see this?
Edit: The proof doesn't make sense since the indexing set $G$ for the union is assumed infinite. Can someone supply a proof or a counterexample?
