# Finding when a relator can be removed from group presentation

There are 4 Tietze transformations and one of them is removal of superfluous relator. Is there a mechanical approach/algorithm for deciding which relators can be removed and which cannot?

A relator $$R_n$$ can be removed if it can be derived from the other ones $$R_1...R_{n-1}$$. In order to check if such derivation is possible you need to check if $$R_n=e$$ in group $$\langle A| R_1...R_{n-1} \rangle$$. This is called word problem and is undecidable (so there is no algorithm).
• I suspect that, even if the original group is trivial, then it is undecidable whether $R_n$ is redundant i.e. whether $\langle A \mid R_1,\ldots,R_{n-1} \rangle$ is trivial - but I am not sure about that. Oct 6 at 9:35