Quotient Structure on Group defined by Congruence equals Quotient Group

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Theorem

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

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

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

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

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


Then $\struct {G / \mathcal R, \circ_\mathcal R}$ is the subgroup $\struct {G / N, \circ_N}$ of the semigroup $\struct {\powerset G, \circ_\mathcal P}$.


Proof

Let $\eqclass x {\mathcal R} \in G / \mathcal R$.

By Congruence Relation on Group induces Normal Subgroup:

$\eqclass x {\mathcal R} = x N$

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:

$y N = \eqclass x {\mathcal R}$

where:

$\eqclass x {\mathcal R}$ is the equivalence class of $y$ under $\mathcal R$
$\mathcal R$ is the equivalence relation defined by $N$.


Hence the result.

$\blacksquare$


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