Right Congruence Class Modulo Subgroup is Right Coset

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

Left Congruence Class Modulo Subgroup is Left Coset

Sources

1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.1$. The quotient sets of a subgroup
1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Theorem $11.1$
1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.9$: Subgroups: Theorem $11$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 42.3$ Another approach to cosets