Honestly, I have been thinking on this problem for hours but couldn't find a way:
Let $G$ is torsion-free group and $X$ is a maximal independent subset, then $G/\langle X\rangle$ is torsion.
I know:
The main problem is to show that any $g+\langle X\rangle\in \frac{G}{\langle X\rangle}$ has a finite order.
$\langle X\rangle=\sum_{x\in X}\langle x\rangle$.
If the group $G$ is torsion-free and $x\in G$, then the equation $nx=y$ has a unique solution in $G$.
Thanks for any hint.