Baer-Specker group

Let $A$ be a non-empty set, and $G$ an abelian group. The set $K$of all functions from $A$ to $G$ is an abelian group, with additiondefined elementwise by $(f+g)(x)=f(x)+g(x)$. The zero element isthe function that sends all elements of $A$ into $0$ of $G$, and thenegative of an element $f$ is a function defined by$(-f)(x)=-(f(x))$.

When $A=\mathbb{N}$, the set of natural numbers, and $G=\mathbb{Z}$,$K$ as defined above is called the Baer-Specker group. Anyelement of $K$, being a function from $\mathbb{N}$ to $\mathbb{Z}$,can be expressed as an infinite sequence $(x_1,x_2,\mathrm{\dots },x_n,\mathrm{\dots })$, and the elementwise addition on $K$ canbe realized as componentwise addition on the sequences:

$$(x_1,x_2,\mathrm{\dots },x_n,\mathrm{\dots })+(y_1,y_2,\mathrm{\dots },y_n,\mathrm{\dots })=(x_1+y_1,x_2+y_2,\mathrm{\dots },x_n+y_n,\mathrm{\dots }).$$

An alternativecharacterization of the Baer-Specker group $K$ is that it can beviewed as the countably infinite direct product of copies of$\mathbb{Z}$:

$$K={\mathbb{Z}}^{\mathbb{N}}\cong {\mathbb{Z}}^{{\mathrm{\aleph }}_0}=\prod _{{\mathrm{\aleph }}_0}\mathbb{Z}.$$

The Baer-Specker group is an important example of a torsion-freeabelian group whose rank is infinite. It is not a free abeliangroup, but any of its countable subgroup is free (abelian).