I'm studying the Fundamental Theorem of finitely generated Abelian group, and it says that the number of factors equal to $\mathbb Z$ (textbook says it is the Betti number of the group) is unique up to isomorphism.
So what is "the number of factors"?
I tried to find through Wikipedia, and it says that it is the number of generators. So if I'm right, the Betti number of Z_6 is 2 (since 1 and 5 are the generators). Then, what is Betti number of Z_360? Should I try all the cases that are relatively prime to 360? Is there any way to get it easier? I really want to fully understand the definition and applications. Thanks for your help :)
Each finitely generated abelian group $G$ is isomorphic to $$\mathbb{Z}^n\oplus\bigoplus_{m\geq2}\mathbb{Z}_m^{k_m}$$ (where all but finitely many of the $k_m$ are $0$) by the fundamental theorem - which says something much more precise, but this is all we'll need. The Betti number is then $n$; the number of summands (or factors) that are isomorphic to $\mathbb{Z}$. Sometimes the $\mathbb{Z}^n$ is called the free part (and the other summands the torsion part), so you're looking for the number of generators of the free part.
Another more technical way to say this is that it's the dimension over $\mathbb{Q}$ of $G\otimes_{\mathbb{Z}}\mathbb{Q}$; because $\mathbb{Z}\otimes_{\mathbb{Z}}\mathbb{Q}\cong\mathbb{Q}$ and $\mathbb{Z}_m\otimes_{\mathbb{Z}}\mathbb{Q}\cong\{0\}$, this agrees with the above calculation.