\begin{definition}[Definition:Congruence Modulo Subgroup/Left Congruence]
Let $G$ be a group.
Let $H$ be a subgroup of $G$.
We can use $H$ to define a relation on $G$ as follows:
:$\RR^l_H := \set {\tuple {x, y} \in G \times G: x^{-1} y \in H}$
This is called '''left congruence modulo $H$'''.
\end{definition}