I'm trying to solve this question in page 74 of Hungerford's book:
A free abelian group is a free group (Section I.9) if and only if it is cyclic.
I have no idea how to proceed, a solution or a hint would be welcome.
This question helped me to see the difference between free abelian groups and free groups, but I still can't solve the problem.
Thanks in advance
EDIT
Following the comments, my attempt of solution is:
($\Rightarrow$)
If $F$ a free abelian group is a free group itself, then it generated by one element, then it's cyclic.
($\Leftarrow$)
Using the universal property:
Let $F$ be the free group which is cyclic on a set $X$ and $i:X\to F$ the inclusion map. If $G$ is a group and $f:X\to G$ a map of sets, then define $\bar f:F\to G$, $\bar fi(x)=f(x)$, this $\bar f$ is well-defined and unique by definition. (I didn't use the fact $F$ is cyclic)
I would like to know if I am right.
Thanks again