Normal Subgroup induced by Congruence Relation defines that Congruence

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Theorem

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

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

Let $\eqclass e \RR$ be the equivalence class of $e$ under $\RR$.

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


Then $\RR$ is the equivalence relation $\RR_N$ defined by $N$.


Proof

Let $\RR_N$ be the equivalence defined by $N$.

Then:

\(\ds x\) \(\RR\) \(\ds y\)
\(\ds \leadsto \ \ \) \(\ds e\) \(\RR\) \(\ds \paren {x^{-1} \circ y}\) $\RR$ is compatible with $\circ$
\(\ds \leadsto \ \ \) \(\ds \paren {e \circ e}\) \(\RR\) \(\ds \paren {x^{-1} \circ y}\) Group properties
\(\ds \leadsto \ \ \) \(\ds x^{-1} \circ y\) \(\in\) \(\ds N\) Definition of $N$


But from Congruence Class Modulo Subgroup is Coset:

$x \mathrel {\RR_N} y \iff x^{-1} \circ y \in N$

Thus:

$\RR = \RR_N$

$\blacksquare$


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