gr.group theory - Is a normal subgroup of a finitely presented group finitely generated or normal finitely genrated?

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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