Free abelian with basis definition

Question

Let $F$ be an abelian group and $B$ a subset of $F$. $F$ is free abelian with basis $B$ if and only if the cyclic subgroup $\langle b\rangle $ is infinite cyclic for each $b\in B$ and $F$ is the direct sum of $\langle b\rangle $.

My question is, $\langle b\rangle $ is already infinite, so why does the author need to say that it is "infinite cyclic", especially given that the author states that it is a cyclic subgroup. Why doesn't the author just say the "cyclic subgroup $\langle b\rangle$ is infinite"?

Is there something I'm missing?

Answer 1

I’m not sure what you mean by “is already infinite”...

Let’s look at the definition, and what we know before we get to that sentence:

Let $F$ be an abelian group and $B$ a subset of it.

So, we know $F$ is abelian, we know $B$ is a subset. That’s it. We don’t know yet if $F$ is finite or infinite, torsion, torsion-free, or mixed, or anything else. Just that $F$ is abelian.

$F$ is free abelian with basis $B$ if and only if the cyclic subgroup $\langle b\rangle$ is infinite cyclic for each $b\in B$, and $F$ is the direct sum of [the] $\langle b\rangle$.

We know $\langle b\rangle$ is cyclic; we do not know it is infinite. Perhaps that’s what you meant? That we know it’s cyclic so why do we need ot say it it is infinite and cyclic? I suspect that this is just a question of the “name” given to these groups. It is not uncommon to refer to/define the subgroup generated by a non-torsion element as “infinite cyclic”, so you should think of that not as two adjectives describing the group (“it is both infinite and it is cyclic”), but rather as a single compound adjective describing it (“it is infinitecyclic”).