Let $A$ be an abelian group with generators $x_1,x_2, \cdots, x_n$ and defining relations conssisting of $[x_i,x_j]$, $i<j=1,2, \cdots, n$, and $r$ further relations. If $r<n$, prove that $A$ is infinite.
Let $F$ be the free abelian group on $n$ generators. Do I have to prove the $r$ relations generate a finite subgroup of $F$? How to?
Thank you very much!