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.


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.


  • $\begingroup$ hi,Steve. Thanks for your answer. Would you like to give some explainations or some references for your arguments? $\endgroup$ – yeshengkui Mar 22 '10 at 6:24
  • 1
    $\begingroup$ 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 $N$>. $\endgroup$ – Steve D Mar 22 '10 at 8:28

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.