Let $(G,\cdot)$ be a group with identity element $e$, and let $H \le G$ be a subgroup. Define a relation $\sim_R$ on $G$ by declaring that, for $x,y \in G$,
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\begin{align*}
x \sim_R y \quad \Longleftrightarrow \quad yx^{-1} \in H.
\end{align*}
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Then $\sim_R$ is an [equivalence relation](/page/Equivalence%20Relation) on $G$. Moreover, for each $x \in G$, the equivalence class of $x$ is the right coset
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\begin{align*}
Hx := \{hx : h \in H\}.
\end{align*}