Equivalent Statements for Congruence Modulo Subgroup/Right

Theorem

Let $G$ be a group.

Let $H$ be a subgroup of $G$.

Let $x \equiv^r y \pmod H$ denote that $x$ is right congruent modulo $H$ to $y$.

Then the following statements are equivalent:

$\text {(1)}: \quad$

$\ds x$

$\equiv^r$

$\ds y \pmod H$

$\text {(2)}: \quad$

$\ds x y^{-1}$

$\in$

$\ds H$

$\text {(3)}: \quad$

$\ds \exists h \in H: \, $

$\ds x y^{-1}$

$=$

$\ds h$

$\text {(4)}: \quad$

$\ds \exists h \in H: \, $

$\ds x$

$=$

$\ds h y$

$\ds x$

$\equiv^r$

$\ds y \pmod H$

$\ds \leadstoandfrom \ \ $

$\ds x y^{-1}$

$\in$

$\ds H$

Definition of Right Congruence Modulo Subgroup

$\ds \leadstoandfrom \ \ $

$\ds \exists h \in H: \, $

$\ds x y^{-1}$

$=$

$\ds h$

Definition of Element of $H$

$\ds \leadstoandfrom \ \ $

$\ds \exists h \in H: \, $

$\ds x$

$=$

$\ds h y$

$\blacksquare$

1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 42.2$ Another approach to cosets