My Professor defines a congruence relation $R$ over a group $G$ as $aRb\wedge cRd\Rightarrow acRbd$, for each $a,b,c,d\in G$.
I'm trying to prove the following: Let $G$ be a group and $H$ is a subgroup of $G$, and consider the equivalence relations $R_d^H$ and $R_e^H$ defined as
$xR_d^Hy\Leftrightarrow xy^{-1}\in H$ and $xR_e^Hy\Leftrightarrow x^{-1}y\in H$.
Then the following statements are equivalent:
1) $R_d^H$ is a congruence relation;
2) $R_d^H=R_e^H$;
3) $R_e^H$ is a congruence relation.
Any tips on how to prove this? Thanks!