Congruence Relation induces Normal Subgroup

Theorem
Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $\RR$ be a congruence relation for $\circ$.

Let $H = \eqclass e \RR$, where $\eqclass e \RR$ is the equivalence class of $e$ under $\RR$.

Then:
 * $(1): \quad \struct {H, \circ \restriction_H}$ is a normal subgroup of $G$


 * $(2): \quad \RR$ is the equivalence relation $\RR_H$ defined by $H$


 * $(3): \quad \struct {G / \RR, \circ_\RR}$ is the subgroup $\struct {G / H, \circ_H}$ of the semigroup $\struct {\powerset G, \circ_\PP}$.

Also see

 * Congruence Modulo Normal Subgroup is Congruence Relation, the converse of this result