Show that $\langle s,t \mid t^2, tst^{-1}s \rangle \cong \langle a,b \mid a^2, b^2 \rangle$.

Question

I´m trying to comprehend why those groups are isomorphic:

$$\langle s,t \mid t^2, tst^{-1}s \rangle \cong \langle a,b \mid a^2, b^2 \rangle.$$

Can anybody help?

Answer 1

Use Tietze transformations.

I'd start by trying to write the relator $tst^{-1}s$ in terms of a new generator $g$ on the LHS presentation then show that it eliminates one of the other generators in such a way that $t^2$ is preserved and $g^2$ holds, with no other (non-trivial) relators.

I'm afraid there's no one-size-fits-all method for these types of questions; indeed, this can be proven to be the case.