<s>[INST] A certain subgroup is isomorphic to Klein group I'm given this presentation
$$G=\langle x,y \mid x^2=1, y^3=1, (xy)^4=1\rangle,$$
and I have to prove that $N=\langle (xy)^2, (yx)^2\rangle$ is isomorphic to the Klein group $V=\{1, a, b, ab\}$.
I've tried identifying $(xy)^2$ with $a$ and $(yx)^2$ with $b$, which until now is ok because $((xy)^2)^2=1, ((yx)^2)^2=1$, but I have no idea how to prove that $((xy)^2(yx)^2)^2=1$
Can anyone help?
 [/INST] We have $(xy)^2 = (xy)^{-2} = (y^{-1}x^{-1})^2 = (y^2x)^2$. And $(yx)^2 = (yx)^{-2}=(x^{-1}y^{-1})^2 = (xy^2)^2$.
So
$$\begin{align*}
\Bigl( (xy)^2(yx)^2\Bigr)^2 &= \Bigl( (y^2x)^2(xy^2)^2\Bigr)^2\\
&= \Bigl( y^2xy^2xxy^2xy^2\Bigr)^2\\
&= \Bigl( y^2xy^4xy^2\Bigr)^2\\
&= \Bigl( y^2xyxy^2\Bigr)^2\\
&= y^2xyxy^2y^2xyxy^2\\
&= y^2xyxyxyxy^2\\
&= y(yx)(yx)(yx)(yx)y^2\\
&= y(yx)^4y^2\\
&= yy^2\\
&=e
\end{align*}$$
</s>