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Let $G$ be a finitely presented group and $N$ a normal subgroup. Is $N$ finitely generated or normally finitely generated? Here normally finitely generation means that for some finite set $S$ of elements in $N$, every element of $N$ can be writen as a product of $G$-conjugation of elements in $S$. Thanks.

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up vote 9 down vote accepted

If $G$ is the free group on two generators, then $N$ the commutator subgroup is not finitely generated. If $H$ is any finitely generated, but not finitely presented group, then $H$ is the quotient of a finitely generated free group $G$, with kernel $N$ which is not normally finitely generated.

Steve

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hi,Steve. Thanks for your answer. Would you like to give some explainations or some references for your arguments? –  yeshengkui Mar 22 '10 at 6:24
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If $H$ is finitely generated, then there exists a finite rank free group $G$ surjecting onto $H$. If the kernel $N$ was finitely normally generated, $H$ would have a finite presentation, given by < generators for $G$ | normal generators for $N$>. –  Steve D Mar 22 '10 at 8:28

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