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If $G\cong \mathbb{Z}\times \mathbb{Z}\times \dots \times \mathbb{Z}$ is a finitely generated abelian group without torsion of rank $n$, where $n$ is the number of copies of $\mathbb{Z}$. Then any subgroup $H$ in $G$ is also finitely generated without torsion of rank $m$ where $m\leq n$

It looks clear, but how can we show this? What are the possible approaches to this question?

  • 1
    See [this](http://math.stackexchange.com/q/107630/742), and [this](http://math.stackexchange.com/a/32807/742).2012-03-07
  • 0
    Okay, this should be a FAQ...2012-03-08

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