Torsion subgroup of a finitely-generated abelian group is finite?

Question

The above claim was made at the very beginning of a proof of the structure theorem for finitely-generated abelian groups and brushed off as easy. However, I think the problem is easy if the torsion subgroup is finitely-generated, but this does not seem to necessarily be true, or at least not obviously. Must the torsion subgroup be finitely-generated? And is the claim in the title true?

Answer 1

Any subgroup of a finitely generated abelian group (not true for non abelian groups) is finitely generated. So yes, if you have a finitely generated abelian group then its torsion subgroup is also finitely generated, and hence must be isomorphic to a group of the form $\mathbb{Z_{m_1}}\times\mathbb{Z_{m_2}}\times...\times\mathbb{Z_{m_k}}\times \mathbb{Z^r}$ when $m_1|m_2|...|m_k$. Now, all the elements of the torsion subgroup must have finite order so $r=0$ and we really get that it is finite.