Quotient Structure on Group defined by Congruence equals Quotient Group

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

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

Let $\struct {G / \RR, \circ_\RR}$ be the quotient structure defined by $\RR$.

Let $N = \eqclass e \RR$ be the normal subgroup induced by $\RR$.

Let $\struct {G / N, \circ_N}$ be the quotient group of $G$ by $N$.

Then $\struct {G / \RR, \circ_\RR}$ is the subgroup $\struct {G / N, \circ_N}$ of the semigroup $\struct {\powerset G, \circ_\PP}$.

Proof
Let $\eqclass x \RR \in G / \RR$.

By Congruence Relation on Group induces Normal Subgroup:
 * $\eqclass x \RR = x N$

where $x N$ is the (left) coset of $N$ in $G$.

Similarly, let $y N \in G / N$.

Then from Normal Subgroup induced by Congruence Relation defines that Congruence:
 * $y N = \eqclass x \RR$

where:
 * $\eqclass x \RR$ is the equivalence class of $y$ under $\RR$
 * $\RR$ is the equivalence relation defined by $N$.

Hence the result.

Also see

 * Congruence Relation induces Normal Subgroup


 * Congruence Relation on Group induces Normal Subgroup
 * Normal Subgroup induced by Congruence Relation defines that Congruence