Left Congruence Class Modulo Subgroup is Left Coset

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Theorem

Let $G$ be a group, and let $H \le G$ be a subgroup.

Let $\RR^l_H$ be the equivalence defined as left congruence modulo $H$.

The equivalence class $\eqclass g {\RR^l_H}$ of an element $g \in G$ is the left coset $g H$.


This is known as the left congruence class of $g \bmod H$.


Proof

Let $x \in \eqclass g {\RR^l_H}$.

Then:

\(\ds x\) \(\in\) \(\ds \eqclass g {\RR^l_H}\)
\(\ds \leadsto \ \ \) \(\ds \exists h \in H: \, \) \(\ds g^{-1} x\) \(=\) \(\ds h\) Definition of Left Congruence Modulo $H$
\(\ds \leadsto \ \ \) \(\ds \exists h \in H: \, \) \(\ds x\) \(=\) \(\ds g h\) Group properties
\(\ds \leadsto \ \ \) \(\ds x\) \(\in\) \(\ds g H\) Definition of Left Coset
\(\ds \leadsto \ \ \) \(\ds \eqclass g {\RR^l_H}\) \(\subseteq\) \(\ds g H\) Definition of Subset


Now let $x \in g H$.

Then:

\(\ds x\) \(\in\) \(\ds g H\)
\(\ds \leadsto \ \ \) \(\ds \exists h \in H: \, \) \(\ds x\) \(=\) \(\ds g h\) Definition of Left Coset
\(\ds \leadsto \ \ \) \(\ds g^{-1} x\) \(=\) \(\ds h \in H\) Group properties
\(\ds \leadsto \ \ \) \(\ds x\) \(\in\) \(\ds \eqclass g {\RR^l_H}\) Definition of Left Congruence Modulo $H$
\(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds g H \subseteq \eqclass g {\RR^l_H}\) Definition of Subset


Thus:

$\eqclass g {\RR^l_H} = g H$

that is, the equivalence class $\eqclass g {\RR^l_H}$ of an element $g \in G$ equals the left coset $g H$.

$\blacksquare$


Also see


Sources

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