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.
Given the Betti number equals the number of elements in some generating set in a finitely generated abelian group $G$.
To show that $G$ is free abelian, I want to show that $G$ is torsion-free.
Let $\{x_1,\dots,x_r\}$ be a generating set of $G$ and $T$ be a torsion group of $G$.
Then $G/T$ is of rank $r$ and is generated by $\{x_1T,\dots,x_rT\}$. But here we are not sure that whether $\{x_1T,\dots,x_rT\}$ forms a basis for $G/T$. Also I still can't get any relevant information to show that $G$ is torsion-free.