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It is well known that the commutator subgroup of a finitely generated nilpotent group is finitely generated, in fact all subgroups in this case are finitely generated.

I am interested in infinite groups in which the commutator subgroup is finitely generated, but not all subgroups are finitely generated. Can someone provide some class of groups which satisfy this? Thanks.

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    $\begingroup$ trivial comment: finitely generated perfect groups and abelian groups. $\endgroup$
    – M.U.
    Oct 7, 2015 at 5:30

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Take any finitely generated nilpotent group $H$, and $A$ to be any abelian group which is not finitely generated group.

Then consider $G=H\times A$. Here $G'=H'$, which is finitely generated, and $A$ is a subgroup of $G$ which is not finitely generated.

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