How to add relations to make groups trivial

Question

Suppose I have a group $G = <x, y|x^2y^5>$, and I want to add a single relation to make G trivial. Is there any way to do this? In general, if we have a group $G = <x, y|r>$, where r is a single relation, is there a good way to find another relation that when added makes the group trivial? Is there an area of study that deals with these "equations"?

Answer 1

You can't do this in general. It would not be possible for example if you took $r = a^{-1}b^{-1}ab$.

But in this example you could for example add $xy^2$. In the group $H = \langle x,y \mid x^2y^5, xy^2 \rangle$, we have $x = y^{-2}$ from the second relator, and hence the first relator becomes just $y$, so $H$ is trivial.