I have the following statement: Every finitely-generated abelian group $G$ is isomorphic to $T\bigoplus F$, where T and F are torsion and free groups.
As an example is given that all abelian groups of order 72 are of this type.
I take all group of order 8: $\mathbb{Z}/8\mathbb{Z}, \mathbb{Z}/2\mathbb{Z}\bigoplus\mathbb{Z}/4\mathbb{Z}, \mathbb{Z}/2\mathbb{Z}\bigoplus\mathbb{Z}/2\mathbb{Z}\bigoplus\mathbb{Z}/2\mathbb{Z}$
And all groups of order 9: $\mathbb{Z}/9\mathbb{Z},\mathbb{Z}/3\mathbb{Z}\bigoplus\mathbb{Z}/3$
but the conclusion is missing.. all groups are torsion groups as finite what about the free part?
If $T$ is torsion, then $T\simeq \{0\}\oplus T$, so the free part is the trivial group. In fact in your definition of free group you have to specify that you want nonzero element to be unique $\mathbb Z$-linear combinations of the basis, and such a condition holds in the trivial group, since it is just the empty condition.