Normal Subgroup induced by Congruence Relation defines that Congruence

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

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

Let $\eqclass e {\mathcal R}$ be the equivalence class of $e$ under $\mathcal R$.

Let $N = \eqclass e {\mathcal R}$ be the normal subgroup induced by $\mathcal R$.

Then $\mathcal R$ is the equivalence relation $\mathcal R_N$ defined by $N$.

Proof
Let $\mathcal R_N$ be the equivalence defined by $N$.

Then:

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

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

Also see

 * Congruence Relation induces Normal Subgroup


 * Congruence Relation on Group induces Normal Subgroup
 * Quotient Structure on Group defined by Congruence equals Quotient Group