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There is a well known fact says that if $H$ is a finite index subgroup of $G$ then $H$ has a finite index subgroup $N$ which is normal in $G$. If $H$ is not normal in $G$ then one can find $N$ proper, namely $N \neq H$. My question is as follows: If $H$ is normal in $G$, is there always a proper finite index subgroup $N$ in $H$ which is normal in $G$?

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    Well, not if $H = \{e\}$, of course. Probably you want $G$ to be infinite?2011-08-01
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    It seems that we don't have the same interpretation of your question. I think that the answer will be “no” in all cases, but it might be worthwhile to rephrase it.2011-08-02

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