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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, and $r$ further relations. If $r, 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!

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    @ShinyaSa$k$ai: yes, your solution is correct.2011-08-10

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