Right Congruence Class Modulo Subgroup is Right Coset

From ProofWiki
Jump to navigation Jump to search

Theorem

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


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

The equivalence class $\eqclass g {\RR^r_H}$ 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 $x \in \eqclass g {\RR^r_H}$.

Then:

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


Now let $x \in g H$.

Then:

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


Thus:

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

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

$\blacksquare$


Also see


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Right_Congruence_Class_Modulo_Subgroup_is_Right_Coset&oldid=565265"