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Let $G$ be a f.g. residually torsion-free nilpotent group. Let $x$ be a nontrivial element of G, then there is a normal subgroup $N$ of $G$ such as $G/N$ torsion-free nilpotent, $x \notin N$. Let $\gamma_{n}(K)$ be the n-term of the lower central series of K. Since $G/N$ is nilpotent, there is $n$ such as $N\gamma_{n}(G)/N = \gamma_{n}(G/N) = 1$. It follows that $\gamma_{n}(G) \le N$. Is it true that $G/ \gamma_{n}(G)$ is torsion free?

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    What is $K$?$ $2012-09-09
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    Thanks for the correction. I hope it's ok now.2012-09-09

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