# Pontrjagin's Lemma and an application

I would appreciate any kind of help on the following issue:

On page 114 of Rotman's "Homological Algebra", exercise 3.4 reads:

1) (Pontrjagin) If an abelian group $A$ is countable, torsion-free and its subgroups of finite rank are free abelian, then $A$ itself is free abelian.

2) Every subgroup of finite rank in $\mathbb{Z}^{\mathbb{N}}$ is free abelian.

The author suggests, for the first question, to proceed by expressing $A$ as the union of a family $(P_{\alpha})_{\alpha < \omega}$ of subgroups, indexed by ordinals, and such that $P_{\alpha}$ is always a direct summand in $P_{\alpha + 1}$ and $P_{\beta} = \cup_{\alpha < \beta} P_{\alpha}$. Then, if $X_{\alpha}$ are complements, it does follow from some earlier argument that $A = \oplus_{\alpha} X_{\alpha}$, which is indeed the case.

My question is: how to apply the idea in this setting? The process may go wrong, i.e. for $A = \mathbb{Z}$, start with $P_0 = 2\mathbb{Z}$ which is obviously free, but this cannot be continued. It's not at all obvious how to find those direct summands.

Even worse, there is the "torsion-free" hypothesis which doesn't make any sense (if $na = 0$ then $(a)$ is not free but surely of finite rank!). The statement appears once again further in the book, rephrased but with the same superfluous attribute there.

As to the second one, I don't have any idea due to the same reason. I confess that I'm totally unfamiliar with the general theory of abelian groups which the author seems to assume from an average reader. In fact there is more than a few such items among the exercises in the book, with references to a monograph of Fuchs here and there, making me wonder if this isn't mere display of erudition at the cost of causing confusion.

In the end, I think it's safe to simply assume that (2) is true, so what I'm asking for is a reference for (1) (which I was unable to find).

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