My textbook (author : fraleigh) says that Fundamental theorem of finitely generated abelian groups
Every finitely generated abelian group $G$ is isomorphic to a direct product of cyclic groups of the form $$\Bbb{Z_{(p_1)^{r_1}}}\times \Bbb{Z_{(p_2)^{r_2}}}\times\dots\times \Bbb{Z_{(p_n)^{r_n}}}\times\underbrace{\Bbb{Z}\times\Bbb{Z}\dots\times\Bbb{Z}}_{\text{r times, r : betti number}}$$ where $p_i$ are primes, not necessarily distinct, and $r_i$ are positive integers. The prime powers $(p_i)^{r_i}$ are unique.
I don't understand the theorem. So I was trying to show examples. So I can apply the theorem to several problems. But I don't know what Betti number say. Can you tell me some examples to understand Betti numbers? Actually when I see the next statements, I cannot start.
Any two finitely generated abelian groups with the same Betti number are isomorphic.
It is false.
Every finite abelian group has a Betti number of 0.
It is true.