I'm trying to solve a problem in my textbook which asks me to identify the groups $G_1 = \langle x,y \space | \space x^3y=y^2x^2=x^2y\rangle$ and $G_2 = \langle x,y \space | \space xy=yx, x^4=y^2 \rangle$ from the given presentations.
For $G_1$, I'm pretty sure I can say that $x^3y=x^2y \implies x^3y(x^2y)^{-1} = e \implies x = e$ (where $e$ is the identity), and then $x^3y=y^2x^2 \implies y = y^2$ as $x=e$ to get $G_1 = \langle x,y \space | \space x=y=e \rangle \cong \{e\}$ (though I'd appreciate it if you could tell me if I have this wrong).
What I'm struggling with is trying to do the same sort of thing for $G_2$ - I can't see any way of getting this into a form where I can see the represented group.
I'd appreciate any help you could offer.
$$\begin{align}G_2 &= \langle x, y | xy = yx, x^4 = y^2 \rangle \\ &= \frac{ \langle x, y | xy = yx \rangle }{\langle \langle x^4y^{-2} \rangle \rangle} \\ &= \frac{\langle x \rangle_{\infty} \times \langle y \rangle_{\infty} }{\langle x^4y^{-2} \rangle} \\ &\cong \frac{\langle x \rangle_{\infty} \times \langle y \rangle_{\infty} }{\langle x^2y^{-1} \rangle} \times C_2 \\ &\cong C_{\infty} \times C_2.\end{align}$$