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:
- $\RR = \RR_N$
$\blacksquare$
Also see
- Congruence Relation on Group induces Normal Subgroup
- Quotient Structure on Group defined by Congruence equals Quotient Group
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Theorem $11.5$