Let $G$ be a finitely generated group with a normal subgroup $H$, such that $H$ and the quotient $G/H$ are finitely presented.
Does it follow that $G$ is finitely presented?
I'm attempting to compile a list of properties for which the property on $G$ is equivalent to there being a normal $H$ such that $G$ and $G/H$ both have that property. I've got solubility and finite generation for which the proofs feel sort of similar but I've tried a similar thing here and not had any luck. Any help much appreciated.
Yes. Let $\langle X \mid R \rangle$ and $\langle Y \mid S \rangle$ be finite presentations of $G/H$ and of $H$ respectively. Let $\hat{X}$ be a set of inverse images of the elements of $G/H$ in $G$.
Then there is a finite presentation $\langle \hat{X} \cup Y \mid \hat{R} \cup S \cup T \rangle$ of $G$, where:
(i) $\hat{R}$ are in one-one correspondence with $R$, and the elements of $\hat{R}$ have the form $\hat{r}w_r^{-1}$, where $\hat{r}$ is the corresponding relator $r \in R$ but with its generators $x \in X$ replaced by the corresponding $\hat{x} \in \hat{X}$, and $w_r$ is a word in $Y \cup Y^{-1}$ which is equal to $\hat{r}$ evaluated as an element of $H$; and
(ii) the elements of $T$ have the form $\hat{x}^{-1}y\hat{x}w_{xy}^{-1} $, one for each $x \in X$ and $y \in Y$, and $w_{xy}$ is a word in $Y \cup Y^{-1}$ equal to $\hat{x}^{-1}y\hat{x}$ evaluated in $H$.