The question is as follows:
Find an example of a group $G$ with a subgroup $H$ so that $$\{(x, y) | xx^{−1} y^{−1} \in H\}$$ is not an equivalence relation on $G$.
I've just been working on this problem set for hours now and I'm having a hard time coming up with an example for this question.
Consider $G=V=\Bbb Z_2\times \Bbb Z_2$ given by the presentation $$\langle a, b\mid a^2, b^2, ab=ba\rangle$$ and the subgroup $H\cong \Bbb Z_2$ given by $\langle b\mid b^2\rangle$. Pick $a\in G\setminus H$. Then $a\not\sim a$.
Another way to see this is that the condition $xx^{-1}y^{-1}\in H$ is equivalent to $ey^{-1}=y^{-1}\in H$, which is in turn equivalent to $y\in H$ since $H$ is a subgroup of $G$. Thus it is sufficient to let $x\in G\setminus H$ in order for $x\not\sim x$; that is, for reflexivity to fail.