We know that any torsion-free group can be imbedded in a vector space over $\mathbb Q$. Now the question is:
if there is a maximal independent subset with $n$ elements of our torsion-free group $G$,$X$, then the group can be imbbeded in a vector space of dimention $n$.
I start with the fact that, we can establish an injective from $G$ to $\mathbb V$, a vector space over $\mathbb Q$ , namely $f$. This function is bijective form $G$ to $f$($G$), so $f$($X$) is independent set in $\mathbb V$. How can I extend this former set into a basis for $\mathbb V$ ?