This is one of the Exercise problem in Hatcher's AT.
(Exercise 3.3.18) Show that a direct limit $\varinjlim G_\alpha$ of torsionfree abelian group $G_\alpha$ is torsionfree. More generally, show that any finitely generated subgroup of $\varinjlim G_\alpha$ is realized as a subgroup of some $G_\alpha$.
I've already proved the first statement (torsionfree). For the second statement, I want to use the torsionfree result but I don't know how to connected these two. There's a post especially for the second statement. But I don't want to use finitely presented concept for this (Hatcher didn't introduce such concept). Could you help?
Best Answer
$G_{\alpha}$ is described in the first sentence as torsion free abelian, and that persists for the second sentence. In the second part you are considering a finitely generated subgroup $H$ of the direct limit, and you know that the direct limit is torsion free by part one. Subgroups of torsion free abelian groups must be torsion free, so the subgroup is finitely generated torsion free abelian, and therefore (for example by the link I posted earlier) must be free abelian. Choose generators $\{a_1, ..., a_n\}$ for the subgroup $H$. Now you could imitate the proof in the post that you cited: each generator $a_i$ must lie in the image (in the direct limit) of some $G_\alpha$, and since there are finitely many of them, they must all lie in the image of a common $G_\alpha$, and therefore the subgroup itself is in the image of that $G_\alpha$.