TITLE: What is the difference between a free Abelian group (of say a finite basis) and a finitely generated Abelian group?
QUESTION [7 upvotes]: Is the difference that the generating set of the finitely generating Abelian group need not be linearly independent? I understand that both the basis and the finite generating set span the Abelian group. Thanks for clarifying.

REPLY [3 votes]: I personally find the following perspective useful. This also generalizes nicely to other situations. 
An abelian group $G$ is free whenever there is an isomorphism 
$$
\alpha\colon\mathbf{Z}^I \to G
$$
for some index set $I$.
The usual generators of $\mathbf{Z}^I$ are then mapped to generators of $G$ by $\alpha$. 
An abelian group $H$ is finitely generated whenever there is a surjective homomorphism
$$
\beta\colon\mathbf{Z}^n \to H. 
$$
for some natural number $n\in\mathbf{N}$. 
Again, $\beta$ maps generators to generators, but now they may be subject to relations. In other words, there is some subgroup $R$ of $\mathbf{Z}^n$ such that $H$ is isomorphic to $\mathbf{Z} ^n / R$.
In the situations above, if $I$ happens to be a finite set, or if $\beta$ happens to be injective (or equivalently, $R=0$), then $G$ or $H$ respectively are both free and finitely generated.