4 (i) (Pontrjagin) If A is a countable torsion-free abelian...

4 (i) (Pontrjagin) If $A$ is a countable torsion-free abelian group each of whose subgroups $S$ of finite rank is free abelian, prove that $A$ is free abelian (the $\operatorname{rank}$ of an abelian group $S$ is defined as $\operatorname{dim}_{\mathbb{Q}}\left(\mathbb{Q} \otimes_{\mathbb{Z}} S\right)$; cf. Exercise $2.36$ on page 97). Hint. See the discussion on page 103 . (ii) Prove that every subgroup of finite rank in $\mathbb{Z}^{\mathbb{N}}$ (the product of countably many copies of $\mathbb{Z}$ ) is free abelian. (iii) Prove that every countable subgroup of $\mathbb{Z}^{\mathbb{N}}$ is free. (In Theorem $4.17$, we will see that $\mathbb{Z}^{\mathbb{N}}$ itself is not free.)

4 (i) (Pontrjagin) If $A$ is a countable torsion-free abelian group each of whose subgroups $S$ of finite rank is free abelian, prove that $A$ is free abelian (the $\operatorname{rank}$ of an abelian group $S$ is defined as $\operatorname{dim}_{\mathbb{Q}}\left(\mathbb{Q} \otimes_{\mathbb{Z}} S\right)$; cf. Exercise $2.36$ on page 97).
Hint. See the discussion on page 103 .
(ii) Prove that every subgroup of finite rank in $\mathbb{Z}^{\mathbb{N}}$ (the product of countably many copies of $\mathbb{Z}$ ) is free abelian.
(iii) Prove that every countable subgroup of $\mathbb{Z}^{\mathbb{N}}$ is free. (In Theorem $4.17$, we will see that $\mathbb{Z}^{\mathbb{N}}$ itself is not free.)

Key Concepts

Torsion?Free Abelian Groups

A torsion?free abelian group is one in which the only element that has finite order is the identity. This property guarantees that when the group is embedded in its rational vector space (via tensoring with the rationals), the structure behaves more like a vector space, which is critical for applying techniques such as finding bases and defining rank.

Free Abelian Groups

A free abelian group is an abelian group that has a basis such that every element can be uniquely expressed as a finite linear combination of basis elements with integer coefficients. This concept is fundamental because free abelian groups are the most well?understood abelian groups, analogous to vector spaces over fields, and establishing that a given group is free often simplifies its study and classification.

Countable Groups

A countable group is one whose elements can be listed in a sequence, meaning the set of elements is either finite or has the same cardinality as the natural numbers. Countability is significant in this context since many structural results and inductive arguments rely crucially on being able to enumerate group elements.

Subgroups of Finite Rank

A subgroup of finite rank is a subgroup for which there exists a finite bound on the number of linearly independent elements over the integers, typically defined using a tensor product with the rational numbers. This concept links abelian group theory with linear algebra, allowing utilization of dimension theory from vector spaces when analyzing subgroups.

Rank Defined via the Tensor Product with the Rationals

The rank of an abelian group is defined as the dimension of the vector space obtained by tensoring the group with the rationals over the integers. This construction turns the group into a rational vector space, giving a precise measure of the 'size' of its free part, and is a vital tool for comparing different abelian groups in terms of their structural complexity.

Direct Product versus Direct Sum in Abelian Groups

The distinction between the direct product and the direct sum is crucial in studying infinite collections of abelian groups. While the direct sum consists of elements with only finitely many nonzero components and is often free when each component is free, the direct product allows infinitely many nonzero entries. This difference is key in understanding why certain subgroups of a direct product may be free even if the entire direct product is not.

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