# Why is the rank of a finitely generated finite abelian group zero?

I haven't yet proven the fundamental theorem of finitely generated abelian groups, but it is stated without proof in my textbook, Abstract Algebra by Dummit and Foote, page 159; its proof appears in a later chapter.

The textbook also states that the theorem implies that a finitely generated abelian group is a finite group if and only if its free rank is zero.

How so?

$$\mathbb{Z}^n \oplus T,$$
with the torsion part $T$ being the subgroup of all elements of finite order. Thus, a f.g. abelian group is finite iff both $n$ and the free rank of the group equal zero.