Normal Subgroup induced by Congruence Relation defines that Congruence

Theorem
Let $\left({G, \circ}\right)$ be a group whose identity is $e$.

Let $\mathcal R$ be a congruence relation for $\circ$.

Let $\left[\!\left[{e}\right]\!\right]_\mathcal R$ be the equivalence class of $e$ under $\mathcal R$.

Let $H = \left[\!\left[{e}\right]\!\right]_\mathcal R$ be the normal subgroup induced by $\mathcal R$.

Then $\mathcal R$ is the equivalence relation $\mathcal R_H$ defined by $H$.

Proof
Let $\mathcal R_H$ be the equivalence defined by $H$.

Then:

But from Congruence Class Modulo Subgroup is Coset:
 * $x \mathop {\mathcal R_H} y \iff x^{-1} \circ y \in H$

Thus:
 * $\mathcal R = \mathcal R_H$

Also see

 * Congruence Relation Gives Rise to Normal Subgroup


 * Congruence Relation on Group induces Normal Subgroup
 * Quotient Structure on Group Defined by Congruence Equals Quotient Group