Left Congruence Class Modulo Subgroup is Left Coset

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Theorem

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

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

The equivalence class $\eqclass g {\mathcal R^l_H}$ 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 $x \in \eqclass g {\mathcal R^l_H}$.

Then:

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


Now let $x \in g H$.

Then:

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


Thus:

$\eqclass g {\mathcal R^l_H} = g H$

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

$\blacksquare$


Also see


Sources