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:

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


Proof

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

$\blacksquare$


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