# 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?

The theorem you haven't proved states that any f.g. abelian group is isomorphic to

$$\mathbb{Z}^n \oplus T,$$

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.

• i have a test about this theorem on monday, and this is the first tims i'm assuming something that i haven't proved. What would be some useful properties this theorem states about finite abelian groups? – user140374 Apr 18 '14 at 22:37
• Well, depending on what version of the theorem you use, or what corollaries can you use: a finite abelian group is a direct sum (when additively written, otherwise "product") of cyclic groups of prime power order and this decomposition is unique up to order of the cyclic factors. Yet I seriously doubt you'll be asked a lot about this theorem if you people haven't yet reachedt he subject, the proof of the theorem and etc. – DonAntonio Apr 18 '14 at 22:43