I'm trying to a build a better understanding of presentations. I get that a a group has a presentation $\langle S \mid R \rangle$ if it is the "freest" group subject to the relations $R$.
But, for instance, how is it immediately clear that $\langle x, y \mid x^2 = y^2 \rangle$ is the same group as $\langle x, y | xyx^{-1}y \rangle$? Also, more generally, what are some ways to determine if two different presentations yield the same group?
Best Answer
As user58512 points out in the comments, it is not always possible to determine whether two presentations define isomorphic groups. It has been shown that the general problem of determining whether a given presentation defines the trivial group is undecidable.
However, in many cases we can show (or even easily see) that two presentations represent the same group. One way to do this formally is using Tietze transformations. I suggest you try transforming your first presentation into the second using Tietze tranformations. As a hint, try starting by adding $xy$ as a new generator in the second presentation.