There is a question on mathoverflow regarding the finitely generated center of a finitely generated group.
In the first remark of the question it is written that "If $G$ is a finitely generated group with infinitely generated center $Z(G)$, then the quotient $G/Z(G)$ is not finitely presented (as follows from a result of B.H Neumann)."
I need the reference of this result; i.e., I want to know from which result of B.H. Neumann the first remark follows. It would be helpful for me. Thank you.
Here is a sketch proof. Let $X$ be a finite generating set of $G$, so $G = F_X/K$ for some $K$, where $F_X$ is the free group on $X$. Then $N = L/K$ for some $L$.
Now if $G/N \cong F_X/L$ has a finite presentation then it has one on the images of $X$ in $G/N$. Let $\langle X \mid R \rangle$ be such a presentation. Then, by definition, $L$ is the normal closure $\langle R^F \rangle$ of $R$ in $F$.
Now $L/K \le Z(F_X/K)$ implies that $N=L/K$ is generated by the finite set $\{rK : r \in R \}$.