Right Congruence Class Modulo Subgroup is Right Coset

From ProofWiki
Jump to navigation Jump to search

Theorem

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


Let $\mathcal R^r_H$ be the equivalence defined as right congruence modulo $H$.

The equivalence class $\eqclass g {\mathcal R^r_H}$ of an element $g \in G$ is the right coset $H g$.


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


Proof

Let $x \in \eqclass g {\mathcal R^r_H}$.

Then:

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


Now let $x \in g H$.

Then:

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


Thus:

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

That is, the equivalence class $\eqclass g {\mathcal R^r_H}$ of an element $g \in G$ equals the right coset $g H$.

$\blacksquare$


Also see


Sources