Equivalent Statements for Congruence Modulo Subgroup/Right

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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:

\(\text {(1)}: \quad\) \(\ds x\) \(\equiv^r\) \(\ds y \pmod H\)
\(\text {(2)}: \quad\) \(\ds x y^{-1}\) \(\in\) \(\ds H\)
\(\text {(3)}: \quad\) \(\ds \exists h \in H: \, \) \(\ds x y^{-1}\) \(=\) \(\ds h\)
\(\text {(4)}: \quad\) \(\ds \exists h \in H: \, \) \(\ds x\) \(=\) \(\ds h y\)


Proof

\(\ds x\) \(\equiv^r\) \(\ds y \pmod H\)
\(\ds \leadstoandfrom \ \ \) \(\ds x y^{-1}\) \(\in\) \(\ds H\) Definition of Right Congruence Modulo Subgroup
\(\ds \leadstoandfrom \ \ \) \(\ds \exists h \in H: \, \) \(\ds x y^{-1}\) \(=\) \(\ds h\) Definition of Element of $H$
\(\ds \leadstoandfrom \ \ \) \(\ds \exists h \in H: \, \) \(\ds x\) \(=\) \(\ds h y\) Division Laws for Groups

$\blacksquare$


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