> [!NOTE] Definition (Congruence Modulo Subgroup) > Let $G$ be a [[Groups|group]]. Let $H$ be a [[Subgroup|subgroup]] of $G.$ > > The following [[Binary Relation|relation]] on $G$ $R^{l}_{H}=\{ (x,y) \in G \times G \mid x^{-1} y \in H \}$is called the left congruence modulo $H.$ > > The following [[Binary Relation|relation]] on $G$ $R^{l}_{H}=\{ (x,y) \in G \times G \mid x y^{-1} \in H \}$is called the right congruence modulo $H.$ **Notation**: In the cases that $G$ is [[Groups|abelian]] or just that $H$ is [[Normal Subgroup|normal]], we have $R_{H}^{l}=R^{r}_{H}=R_{H}.$ # Properties Note that [[Congruence Modulo Subgroup is Equivalence Relation]].