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Suppose that $G$ is an abelian group and that I want to prove that a certain subgroup $K$ of $G$ has limited exponent and that there exists a normal subgroup $N$ of $G$ with limited exponent. Can I suppose without loss of generality that $N=\{1\}$ and to try to prove that $K$ has limited exponent?

Remark:It is not in my hypothesis that KN/N has exponent.

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    Ops!Remove the word it supposes in begin of the text.2012-04-29

2 Answers 2

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What Arturo was pointing out in his comment was that every subgroup of an abelian group is normal, so if you can prove that $K$ has bounded exponent, then you've found a non-trivial normal subgroup with bounded exponent. You can just let $N = K$.

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Yes, you can, if I am understanding your question correctly. If $N$ has exponent $n$ and $KN/N$ has exponent $m$, then for any $k\in K$, $k^m\in N$, so $k^{mn}=(k^m)^n=1$