Quotient Structure on Group defined by Congruence equals Quotient Group

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

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

Let $\left({G / \mathcal R, \circ_\mathcal R}\right)$ be the quotient structure defined by $\mathcal R$.

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

Let $\left({G / H, \circ_H}\right)$ be the quotient group of $G$ by $H$.

Then $\left({G / \mathcal R, \circ_\mathcal R}\right)$ is the subgroup $\left({G / H, \circ_H}\right)$ of the semigroup $\left({\mathcal P \left({G}\right), \circ_\mathcal P}\right)$.

Also see

 * Congruence Relation on Group induces Normal Subgroup
 * Normal Subgroup Induced by Congruence Relation Defines That Congruence