Computing a presentation for the normal closure

Question

This question may be too general; if so, please let me know and I can try to make it more specific! Given a subgroup of $G$, $H$, with presentation $\langle S\mid R\rangle$, how do we find the presentation of the normal closure of $H$?

Answer 1

There is no algorithm which will give you a "meaningful" output in general, even if we assume that $G$ is free. This is because non-trivial normal subgroups of free groups are finitely generated if and only if they have finite index (see here), and because it is undecidable if a finite presentation $\langle \mathbf{x} \mid\mathbf{r}\rangle$ defines a finite group (as "being finite" is a Markov property). Therefore, in the free group $F(\mathbf{x})$ we cannot determine whether it not the normal closure of the set $\mathbf{r}$ has finite index or not, and hence if it is finitely generated or not.

I used the word "meaningful" above as there may exist an algorithm which will output a description of some generating set of the normal closure of $\mathbf{r}$, but then by the above it is impossible to use this description to find a basis for the subgroup. So I think the above captures the essence of the question :-)