Quotient group

Let $G$ be a group, and $R$ an equivalence relation compatible with the group structure on $G$. The structure derived from $R$ on the quotient set $G/R$ is called the quotient group of $G$ by $R$, or the quotient group $G/R$, or $G$ mod $R$.

An equivalence relation $R(x,y)$ on $G$ is compatible with the group structure on $G$ if and only if it is equivalent to a relation of the form $xy^{-1} \in H$, for some normal subgroup $H$ of $G$.

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