Example

Let $G$ be a group and let $H$ be a subgroup of $G$. Then $H$ operates freely by (left or right) translations on $G$.

Let $G$ be a group operating freely on a set $X$, let $R$ be the equivalence relation defined by $G$ on $X$, and let $C \subset X \times X$ be the graph of $R$. If $(x, y) \in C$, then there exists $s \in G$ such that $s.x = y$; and $s$ is unique, since $s.x = s'.x$ implies ${s'}^{-1}s.x = x$, and therefore ${s'}^{-1}s = e$ (as $G$ operates freely). If we make correspond to $(x, y) \in C$ the unique $s \in G$ such that $s.x = y$, we define a mapping $\varphi : C \to G$, which we shall call the canonical mapping of $C$ into $G$. With this notation: